Chapter 5: Exponents and Powers
Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.
What this chapter covers
This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.
5.1 Base
Base is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Base: Explain Base in one simple sentence.
- Look at the words in “Base”.
- State what the idea is used for.
- Give one small mathematical example.
Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.
Answer: Base is a Grade 8 math idea used to describe or solve a specific mathematical relationship.
Worked Example 2
Problem: Base: A student says, “I can use Base without checking units or labels.” Is that a good method?
- Read the statement.
- Ask whether units, signs, labels, or conditions matter.
- Decide whether the method is safe.
Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.
Answer: No. Important units, labels, signs, and conditions must be checked.
Worked Example 3
Problem: Base: What is the first step when solving a problem about Base?
- Read the whole question.
- Underline the known information.
- Identify what must be found.
Very beginner explanation: This prevents you from calculating before you understand the problem.
Answer: Identify the given information and the unknown before choosing a rule.
Worked Example 4
Problem: Base: After solving a Base problem, what should you do before accepting the answer?
- Estimate the expected size or direction.
- Check units and signs.
- Use an inverse method if possible.
Very beginner explanation: Checking is part of solving, not an optional extra.
Answer: Check whether the answer is reasonable and consistent with the problem.
Worked Example 5
Problem: Base: Give one way Base could appear outside a textbook.
- Think about money, measurements, data, maps, geometry, or technology.
- Connect the topic to one of those situations.
Very beginner explanation: Connecting math to real situations makes the rule easier to remember.
Answer: Base can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.
Worked Example 6
Problem: Base: Which representation could help explain Base: a table, graph, diagram, equation, or number line?
- Choose the representation that makes the relationship easiest to see.
- Label it clearly.
Very beginner explanation: Different representations show different features of the same mathematics.
Answer: Use the representation that best matches the problem; more than one may be valid.
Worked Example 7
Problem: Base: A student gets an answer for Base but cannot explain the steps. What should be improved?
- Rewrite the solution one step at a time.
- Name the rule used at each important step.
Very beginner explanation: A correct final number without reasoning may hide an error.
Answer: The reasoning should be shown so the solution can be checked.
Worked Example 8
Problem: Base: Why can estimation help before a detailed Base calculation?
- Round or use benchmark values.
- Predict the approximate answer.
- Compare the exact result with the estimate.
Very beginner explanation: If the exact answer is far from the estimate, recheck the work.
Answer: Estimation gives a target range for the final answer.
Worked Example 9
Problem: Base: How can you test whether your rule for Base works?
- Choose a very small easy example.
- Apply the rule.
- Check the result another way.
Very beginner explanation: Small examples expose mistakes quickly.
Answer: Test the rule on a simple case whose answer can be verified.
Worked Example 10
Problem: Base: How would you teach Base to someone seeing it for the first time?
- Define the idea.
- Show one easy example.
- Explain every step.
- Then let the learner try a similar problem.
Very beginner explanation: This sequence reduces memorization without understanding.
Answer: Teach meaning first, then a small worked example, then guided practice.
Practice exercise
Create one new Grade 8 problem involving Base. Show the important steps, include units when needed, and explain how you checked the answer.
5.2 Exponent
An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Exponent.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Exponent. Show the important steps, include units when needed, and explain how you checked the answer.
5.3 Power
Power is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Power. Show the important steps, include units when needed, and explain how you checked the answer.
5.4 Repeated multiplication
Repeated multiplication is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: An item costs $50.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 50.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $6.50
Worked Example 2
Problem: An item costs $100.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 100.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $13.00
Worked Example 3
Problem: An item costs $75.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 75.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $9.75
Worked Example 4
Problem: An item costs $120.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 120.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $15.60
Worked Example 5
Problem: An item costs $200.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 200.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $26.00
Worked Example 6
Problem: An item costs $150.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 150.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $19.50
Worked Example 7
Problem: An item costs $250.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 250.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $32.50
Worked Example 8
Problem: An item costs $80.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 80.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $10.40
Worked Example 9
Problem: An item costs $300.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 300.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $39.00
Worked Example 10
Problem: An item costs $60.00. Find 13% sales tax.
- Convert 13% to 0.13.
- Tax = 60.00×0.13.
Very beginner explanation: Tax is a percentage of the purchase price and is added to the price when required.
Answer: $7.80
Practice exercise
Create one new Grade 8 problem involving Repeated multiplication. Show the important steps, include units when needed, and explain how you checked the answer.
5.5 Evaluating powers
Evaluating powers is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Evaluating powers. Show the important steps, include units when needed, and explain how you checked the answer.
5.6 Powers of 10
Powers of 10 is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Write 4.5e+06 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 4.5 × 10^6
Worked Example 2
Problem: Write 72000 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 7.2 × 10^4
Worked Example 3
Problem: Write 0.0032 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 3.2 × 10^-3
Worked Example 4
Problem: Write 9.8e+08 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 9.8 × 10^8
Worked Example 5
Problem: Write 4.5e-05 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 4.5 × 10^-5
Worked Example 6
Problem: Write 6.25e+09 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 6.25 × 10^9
Worked Example 7
Problem: Write 8.1e-07 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 8.1 × 10^-7
Worked Example 8
Problem: Write 35700 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 3.57 × 10^4
Worked Example 9
Problem: Write 0.092 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 9.2 × 10^-2
Worked Example 10
Problem: Write 1.2e+06 in scientific notation.
- Move the decimal so exactly one non-zero digit is before it.
- Count how many places the decimal moved.
- Moving left gives a positive exponent; moving right gives a negative exponent.
Very beginner explanation: Scientific notation is a compact way to represent extremely large or small numbers.
Answer: 1.2 × 10^6
Practice exercise
Create one new Grade 8 problem involving Powers of 10. Show the important steps, include units when needed, and explain how you checked the answer.
5.7 Exponent patterns
An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Exponent patterns.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Exponent patterns. Show the important steps, include units when needed, and explain how you checked the answer.
5.8 Multiplying powers with the same base
Multiplying powers with the same base is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Multiplying powers with the same base. Show the important steps, include units when needed, and explain how you checked the answer.
5.9 Dividing powers with the same base
Dividing powers with the same base is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Dividing powers with the same base. Show the important steps, include units when needed, and explain how you checked the answer.
5.10 Power of a power
Power of a power is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Power of a power. Show the important steps, include units when needed, and explain how you checked the answer.
5.11 Zero exponent
An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Zero exponent.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 2
Problem: Evaluate 3^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 3
Problem: Evaluate 4^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 4
Problem: Evaluate 5^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 5
Problem: Evaluate 6^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 6
Problem: Evaluate 7^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 7
Problem: Evaluate 8^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 8
Problem: Evaluate 9^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 9
Problem: Evaluate 10^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Worked Example 10
Problem: Evaluate 12^0.
- Any nonzero base raised to the zero power equals 1.
Very beginner explanation: The zero-exponent rule follows from dividing equal powers of the same nonzero base.
Answer: 1
Practice exercise
Create one new Grade 8 problem involving Zero exponent. Show the important steps, include units when needed, and explain how you checked the answer.
5.12 Negative exponent introduction
An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Negative exponent introduction.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^-1.
- A negative exponent means use the reciprocal.
- 2^-1 = 1/2.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/2
Worked Example 2
Problem: Evaluate 3^-1.
- A negative exponent means use the reciprocal.
- 3^-1 = 1/3.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/3
Worked Example 3
Problem: Evaluate 4^-1.
- A negative exponent means use the reciprocal.
- 4^-1 = 1/4.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/4
Worked Example 4
Problem: Evaluate 5^-1.
- A negative exponent means use the reciprocal.
- 5^-1 = 1/5.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/5
Worked Example 5
Problem: Evaluate 6^-1.
- A negative exponent means use the reciprocal.
- 6^-1 = 1/6.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/6
Worked Example 6
Problem: Evaluate 7^-1.
- A negative exponent means use the reciprocal.
- 7^-1 = 1/7.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/7
Worked Example 7
Problem: Evaluate 8^-1.
- A negative exponent means use the reciprocal.
- 8^-1 = 1/8.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/8
Worked Example 8
Problem: Evaluate 9^-1.
- A negative exponent means use the reciprocal.
- 9^-1 = 1/9.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/9
Worked Example 9
Problem: Evaluate 10^-1.
- A negative exponent means use the reciprocal.
- 10^-1 = 1/10.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/10
Worked Example 10
Problem: Evaluate 12^-1.
- A negative exponent means use the reciprocal.
- 12^-1 = 1/12.
Very beginner explanation: A negative exponent does not make the number negative; it moves the power to the denominator.
Answer: 1/12
Practice exercise
Create one new Grade 8 problem involving Negative exponent introduction. Show the important steps, include units when needed, and explain how you checked the answer.
5.13 Real-life uses of exponents
An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Real-life uses of exponents.
For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.
10 Worked Examples with Answers and Very-Beginner Explanations
Worked Example 1
Problem: Evaluate 2^2.
- Write 2 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 4
Worked Example 2
Problem: Evaluate 3^3.
- Write 3 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 27
Worked Example 3
Problem: Evaluate 4^2.
- Write 4 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 16
Worked Example 4
Problem: Evaluate 5^3.
- Write 5 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 125
Worked Example 5
Problem: Evaluate 6^2.
- Write 6 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 36
Worked Example 6
Problem: Evaluate 7^3.
- Write 7 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 343
Worked Example 7
Problem: Evaluate 8^2.
- Write 8 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 64
Worked Example 8
Problem: Evaluate 9^3.
- Write 9 as a factor 3 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 729
Worked Example 9
Problem: Evaluate 10^2.
- Write 10 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 100
Worked Example 10
Problem: Evaluate 12^2.
- Write 12 as a factor 2 times.
- Multiply the factors.
Very beginner explanation: An exponent tells how many times the base is used as a factor.
Answer: 144
Practice exercise
Create one new Grade 8 problem involving Real-life uses of exponents. Show the important steps, include units when needed, and explain how you checked the answer.
Chapter 5 Review Questions and Answers
Q1. What is important to remember about Base?
Answer: Base is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q2. What is important to remember about Exponent?
Answer: An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Exponent.
Q3. What is important to remember about Power?
Answer: Power is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q4. What is important to remember about Repeated multiplication?
Answer: Repeated multiplication is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q5. What is important to remember about Evaluating powers?
Answer: Evaluating powers is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q6. What is important to remember about Powers of 10?
Answer: Powers of 10 is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q7. What is important to remember about Exponent patterns?
Answer: An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Exponent patterns.
Q8. What is important to remember about Multiplying powers with the same base?
Answer: Multiplying powers with the same base is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q9. What is important to remember about Dividing powers with the same base?
Answer: Dividing powers with the same base is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q10. What is important to remember about Power of a power?
Answer: Power of a power is an important Grade 8 concept in Exponents and Powers. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.
Q11. What is important to remember about Zero exponent?
Answer: An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Zero exponent.
Q12. What is important to remember about Negative exponent introduction?
Answer: An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Negative exponent introduction.
Q13. What is important to remember about Real-life uses of exponents?
Answer: An exponent (a raised number showing repeated multiplication) tells how many times the base is used as a factor. In this section, the focus is Real-life uses of exponents.
Q14. Why should you show your steps?
Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.
Q15. Why is estimation useful?
Answer: Estimation helps you judge whether a calculated answer is reasonable.
Q16. Why do units matter?
Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.
Q17. When should you round?
Answer: Usually round near the end unless the question specifically asks for earlier rounding.
Q18. How can you check an answer?
Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.
Q19. What should you identify first in a word problem?
Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.
Q20. What makes a final answer complete?
Answer: Give the value, correct units when needed, and a brief interpretation in context.
Q21. What should you do if an answer seems unreasonable?
Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.
Q22. Why are diagrams useful?
Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.
Q23. Why are tables useful?
Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.
Q24. Why should you explain your reasoning?
Answer: Reasoning shows why a method works, not just what answer was obtained.
Q25. How do mistakes help learning?
Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.
Q26. When is a calculator useful?
Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.
Q27. Why compare more than one strategy?
Answer: Different strategies may make a problem easier and provide a way to verify the result.
Q28. What is mathematical communication?
Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.
Q29. What is mathematical modelling?
Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.
Q30. Why should assumptions be stated?
Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.