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Chapter 7: Squares, Perfect Squares, and Square Roots

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.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 12 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.

7.1 Square numbers to 144

Square numbers to 144 is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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.

  1. Write 2 as a factor 2 times.
  2. 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.

  1. Write 3 as a factor 3 times.
  2. 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.

  1. Write 4 as a factor 2 times.
  2. 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.

  1. Write 5 as a factor 3 times.
  2. 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.

  1. Write 6 as a factor 2 times.
  2. 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.

  1. Write 7 as a factor 3 times.
  2. 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.

  1. Write 8 as a factor 2 times.
  2. 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.

  1. Write 9 as a factor 3 times.
  2. 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.

  1. Write 10 as a factor 2 times.
  2. 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.

  1. Write 12 as a factor 2 times.
  2. 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 Square numbers to 144. Show the important steps, include units when needed, and explain how you checked the answer.

7.2 Perfect squares

Perfect squares is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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.

  1. Write 2 as a factor 2 times.
  2. 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.

  1. Write 3 as a factor 3 times.
  2. 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.

  1. Write 4 as a factor 2 times.
  2. 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.

  1. Write 5 as a factor 3 times.
  2. 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.

  1. Write 6 as a factor 2 times.
  2. 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.

  1. Write 7 as a factor 3 times.
  2. 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.

  1. Write 8 as a factor 2 times.
  2. 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.

  1. Write 9 as a factor 3 times.
  2. 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.

  1. Write 10 as a factor 2 times.
  2. 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.

  1. Write 12 as a factor 2 times.
  2. 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 Perfect squares. Show the important steps, include units when needed, and explain how you checked the answer.

7.3 Square roots

A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Square roots.

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: Find √4.

  1. Ask which positive number multiplied by itself equals 4.
  2. 2 × 2 = 4.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 2

Worked Example 2

Problem: Find √9.

  1. Ask which positive number multiplied by itself equals 9.
  2. 3 × 3 = 9.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 3

Worked Example 3

Problem: Find √16.

  1. Ask which positive number multiplied by itself equals 16.
  2. 4 × 4 = 16.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 4

Worked Example 4

Problem: Find √25.

  1. Ask which positive number multiplied by itself equals 25.
  2. 5 × 5 = 25.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 5

Worked Example 5

Problem: Find √36.

  1. Ask which positive number multiplied by itself equals 36.
  2. 6 × 6 = 36.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 6

Worked Example 6

Problem: Find √49.

  1. Ask which positive number multiplied by itself equals 49.
  2. 7 × 7 = 49.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 7

Worked Example 7

Problem: Find √64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 8

Worked Example 8

Problem: Find √81.

  1. Ask which positive number multiplied by itself equals 81.
  2. 9 × 9 = 81.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 9

Worked Example 9

Problem: Find √100.

  1. Ask which positive number multiplied by itself equals 100.
  2. 10 × 10 = 100.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 10

Worked Example 10

Problem: Find √144.

  1. Ask which positive number multiplied by itself equals 144.
  2. 12 × 12 = 144.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 12

Practice exercise

Create one new Grade 8 problem involving Square roots. Show the important steps, include units when needed, and explain how you checked the answer.

7.4 Square-root notation

Square-root notation is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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: Find the supplementary angle to 35°.

  1. Supplementary angles total 180°.
  2. 180° - 35° = 145°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 145°

Worked Example 2

Problem: Find the supplementary angle to 48°.

  1. Supplementary angles total 180°.
  2. 180° - 48° = 132°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 132°

Worked Example 3

Problem: Find the supplementary angle to 67°.

  1. Supplementary angles total 180°.
  2. 180° - 67° = 113°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 113°

Worked Example 4

Problem: Find the supplementary angle to 72°.

  1. Supplementary angles total 180°.
  2. 180° - 72° = 108°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 108°

Worked Example 5

Problem: Find the supplementary angle to 110°.

  1. Supplementary angles total 180°.
  2. 180° - 110° = 70°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 70°

Worked Example 6

Problem: Find the supplementary angle to 25°.

  1. Supplementary angles total 180°.
  2. 180° - 25° = 155°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 155°

Worked Example 7

Problem: Find the supplementary angle to 58°.

  1. Supplementary angles total 180°.
  2. 180° - 58° = 122°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 122°

Worked Example 8

Problem: Find the supplementary angle to 83°.

  1. Supplementary angles total 180°.
  2. 180° - 83° = 97°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 97°

Worked Example 9

Problem: Find the supplementary angle to 95°.

  1. Supplementary angles total 180°.
  2. 180° - 95° = 85°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 85°

Worked Example 10

Problem: Find the supplementary angle to 120°.

  1. Supplementary angles total 180°.
  2. 180° - 120° = 60°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 60°

Practice exercise

Create one new Grade 8 problem involving Square-root notation. Show the important steps, include units when needed, and explain how you checked the answer.

7.5 Memorizing squares from 1² to 12²

Memorizing squares from 1² to 12² is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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: Find the supplementary angle to 35°.

  1. Supplementary angles total 180°.
  2. 180° - 35° = 145°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 145°

Worked Example 2

Problem: Find the supplementary angle to 48°.

  1. Supplementary angles total 180°.
  2. 180° - 48° = 132°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 132°

Worked Example 3

Problem: Find the supplementary angle to 67°.

  1. Supplementary angles total 180°.
  2. 180° - 67° = 113°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 113°

Worked Example 4

Problem: Find the supplementary angle to 72°.

  1. Supplementary angles total 180°.
  2. 180° - 72° = 108°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 108°

Worked Example 5

Problem: Find the supplementary angle to 110°.

  1. Supplementary angles total 180°.
  2. 180° - 110° = 70°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 70°

Worked Example 6

Problem: Find the supplementary angle to 25°.

  1. Supplementary angles total 180°.
  2. 180° - 25° = 155°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 155°

Worked Example 7

Problem: Find the supplementary angle to 58°.

  1. Supplementary angles total 180°.
  2. 180° - 58° = 122°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 122°

Worked Example 8

Problem: Find the supplementary angle to 83°.

  1. Supplementary angles total 180°.
  2. 180° - 83° = 97°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 97°

Worked Example 9

Problem: Find the supplementary angle to 95°.

  1. Supplementary angles total 180°.
  2. 180° - 95° = 85°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 85°

Worked Example 10

Problem: Find the supplementary angle to 120°.

  1. Supplementary angles total 180°.
  2. 180° - 120° = 60°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 60°

Practice exercise

Create one new Grade 8 problem involving Memorizing squares from 1² to 12². Show the important steps, include units when needed, and explain how you checked the answer.

7.6 Finding exact square roots

A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Finding exact square roots.

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: Find √4.

  1. Ask which positive number multiplied by itself equals 4.
  2. 2 × 2 = 4.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 2

Worked Example 2

Problem: Find √9.

  1. Ask which positive number multiplied by itself equals 9.
  2. 3 × 3 = 9.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 3

Worked Example 3

Problem: Find √16.

  1. Ask which positive number multiplied by itself equals 16.
  2. 4 × 4 = 16.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 4

Worked Example 4

Problem: Find √25.

  1. Ask which positive number multiplied by itself equals 25.
  2. 5 × 5 = 25.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 5

Worked Example 5

Problem: Find √36.

  1. Ask which positive number multiplied by itself equals 36.
  2. 6 × 6 = 36.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 6

Worked Example 6

Problem: Find √49.

  1. Ask which positive number multiplied by itself equals 49.
  2. 7 × 7 = 49.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 7

Worked Example 7

Problem: Find √64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 8

Worked Example 8

Problem: Find √81.

  1. Ask which positive number multiplied by itself equals 81.
  2. 9 × 9 = 81.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 9

Worked Example 9

Problem: Find √100.

  1. Ask which positive number multiplied by itself equals 100.
  2. 10 × 10 = 100.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 10

Worked Example 10

Problem: Find √144.

  1. Ask which positive number multiplied by itself equals 144.
  2. 12 × 12 = 144.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 12

Practice exercise

Create one new Grade 8 problem involving Finding exact square roots. Show the important steps, include units when needed, and explain how you checked the answer.

7.7 Estimating non-perfect square roots

A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Estimating non-perfect square roots.

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: Find √4.

  1. Ask which positive number multiplied by itself equals 4.
  2. 2 × 2 = 4.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 2

Worked Example 2

Problem: Find √9.

  1. Ask which positive number multiplied by itself equals 9.
  2. 3 × 3 = 9.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 3

Worked Example 3

Problem: Find √16.

  1. Ask which positive number multiplied by itself equals 16.
  2. 4 × 4 = 16.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 4

Worked Example 4

Problem: Find √25.

  1. Ask which positive number multiplied by itself equals 25.
  2. 5 × 5 = 25.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 5

Worked Example 5

Problem: Find √36.

  1. Ask which positive number multiplied by itself equals 36.
  2. 6 × 6 = 36.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 6

Worked Example 6

Problem: Find √49.

  1. Ask which positive number multiplied by itself equals 49.
  2. 7 × 7 = 49.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 7

Worked Example 7

Problem: Find √64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 8

Worked Example 8

Problem: Find √81.

  1. Ask which positive number multiplied by itself equals 81.
  2. 9 × 9 = 81.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 9

Worked Example 9

Problem: Find √100.

  1. Ask which positive number multiplied by itself equals 100.
  2. 10 × 10 = 100.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 10

Worked Example 10

Problem: Find √144.

  1. Ask which positive number multiplied by itself equals 144.
  2. 12 × 12 = 144.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 12

Practice exercise

Create one new Grade 8 problem involving Estimating non-perfect square roots. Show the important steps, include units when needed, and explain how you checked the answer.

7.8 Square roots on a number line

A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Square roots on a number line.

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: Find √4.

  1. Ask which positive number multiplied by itself equals 4.
  2. 2 × 2 = 4.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 2

Worked Example 2

Problem: Find √9.

  1. Ask which positive number multiplied by itself equals 9.
  2. 3 × 3 = 9.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 3

Worked Example 3

Problem: Find √16.

  1. Ask which positive number multiplied by itself equals 16.
  2. 4 × 4 = 16.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 4

Worked Example 4

Problem: Find √25.

  1. Ask which positive number multiplied by itself equals 25.
  2. 5 × 5 = 25.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 5

Worked Example 5

Problem: Find √36.

  1. Ask which positive number multiplied by itself equals 36.
  2. 6 × 6 = 36.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 6

Worked Example 6

Problem: Find √49.

  1. Ask which positive number multiplied by itself equals 49.
  2. 7 × 7 = 49.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 7

Worked Example 7

Problem: Find √64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 8

Worked Example 8

Problem: Find √81.

  1. Ask which positive number multiplied by itself equals 81.
  2. 9 × 9 = 81.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 9

Worked Example 9

Problem: Find √100.

  1. Ask which positive number multiplied by itself equals 100.
  2. 10 × 10 = 100.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 10

Worked Example 10

Problem: Find √144.

  1. Ask which positive number multiplied by itself equals 144.
  2. 12 × 12 = 144.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 12

Practice exercise

Create one new Grade 8 problem involving Square roots on a number line. Show the important steps, include units when needed, and explain how you checked the answer.

7.9 Comparing square roots

A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Comparing square roots.

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: Find √4.

  1. Ask which positive number multiplied by itself equals 4.
  2. 2 × 2 = 4.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 2

Worked Example 2

Problem: Find √9.

  1. Ask which positive number multiplied by itself equals 9.
  2. 3 × 3 = 9.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 3

Worked Example 3

Problem: Find √16.

  1. Ask which positive number multiplied by itself equals 16.
  2. 4 × 4 = 16.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 4

Worked Example 4

Problem: Find √25.

  1. Ask which positive number multiplied by itself equals 25.
  2. 5 × 5 = 25.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 5

Worked Example 5

Problem: Find √36.

  1. Ask which positive number multiplied by itself equals 36.
  2. 6 × 6 = 36.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 6

Worked Example 6

Problem: Find √49.

  1. Ask which positive number multiplied by itself equals 49.
  2. 7 × 7 = 49.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 7

Worked Example 7

Problem: Find √64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 8

Worked Example 8

Problem: Find √81.

  1. Ask which positive number multiplied by itself equals 81.
  2. 9 × 9 = 81.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 9

Worked Example 9

Problem: Find √100.

  1. Ask which positive number multiplied by itself equals 100.
  2. 10 × 10 = 100.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 10

Worked Example 10

Problem: Find √144.

  1. Ask which positive number multiplied by itself equals 144.
  2. 12 × 12 = 144.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 12

Practice exercise

Create one new Grade 8 problem involving Comparing square roots. Show the important steps, include units when needed, and explain how you checked the answer.

7.10 Area and square roots

A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Area and square roots.

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: Find √4.

  1. Ask which positive number multiplied by itself equals 4.
  2. 2 × 2 = 4.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 2

Worked Example 2

Problem: Find √9.

  1. Ask which positive number multiplied by itself equals 9.
  2. 3 × 3 = 9.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 3

Worked Example 3

Problem: Find √16.

  1. Ask which positive number multiplied by itself equals 16.
  2. 4 × 4 = 16.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 4

Worked Example 4

Problem: Find √25.

  1. Ask which positive number multiplied by itself equals 25.
  2. 5 × 5 = 25.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 5

Worked Example 5

Problem: Find √36.

  1. Ask which positive number multiplied by itself equals 36.
  2. 6 × 6 = 36.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 6

Worked Example 6

Problem: Find √49.

  1. Ask which positive number multiplied by itself equals 49.
  2. 7 × 7 = 49.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 7

Worked Example 7

Problem: Find √64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 8

Worked Example 8

Problem: Find √81.

  1. Ask which positive number multiplied by itself equals 81.
  2. 9 × 9 = 81.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 9

Worked Example 9

Problem: Find √100.

  1. Ask which positive number multiplied by itself equals 100.
  2. 10 × 10 = 100.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 10

Worked Example 10

Problem: Find √144.

  1. Ask which positive number multiplied by itself equals 144.
  2. 12 × 12 = 144.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 12

Practice exercise

Create one new Grade 8 problem involving Area and square roots. Show the important steps, include units when needed, and explain how you checked the answer.

7.11 Calculator use

Calculator use is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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: Calculator use: Explain Calculator use in one simple sentence.

  1. Look at the words in “Calculator use”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Calculator use is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Calculator use: A student says, “I can use Calculator use without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. 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: Calculator use: What is the first step when solving a problem about Calculator use?

  1. Read the whole question.
  2. Underline the known information.
  3. 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: Calculator use: After solving a Calculator use problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. 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: Calculator use: Give one way Calculator use could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Calculator use can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Calculator use: Which representation could help explain Calculator use: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. 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: Calculator use: A student gets an answer for Calculator use but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. 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: Calculator use: Why can estimation help before a detailed Calculator use calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. 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: Calculator use: How can you test whether your rule for Calculator use works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. 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: Calculator use: How would you teach Calculator use to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. 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 Calculator use. Show the important steps, include units when needed, and explain how you checked the answer.

7.12 Real-life square-root problems

Real-life square-root problems is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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: Find the supplementary angle to 35°.

  1. Supplementary angles total 180°.
  2. 180° - 35° = 145°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 145°

Worked Example 2

Problem: Find the supplementary angle to 48°.

  1. Supplementary angles total 180°.
  2. 180° - 48° = 132°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 132°

Worked Example 3

Problem: Find the supplementary angle to 67°.

  1. Supplementary angles total 180°.
  2. 180° - 67° = 113°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 113°

Worked Example 4

Problem: Find the supplementary angle to 72°.

  1. Supplementary angles total 180°.
  2. 180° - 72° = 108°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 108°

Worked Example 5

Problem: Find the supplementary angle to 110°.

  1. Supplementary angles total 180°.
  2. 180° - 110° = 70°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 70°

Worked Example 6

Problem: Find the supplementary angle to 25°.

  1. Supplementary angles total 180°.
  2. 180° - 25° = 155°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 155°

Worked Example 7

Problem: Find the supplementary angle to 58°.

  1. Supplementary angles total 180°.
  2. 180° - 58° = 122°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 122°

Worked Example 8

Problem: Find the supplementary angle to 83°.

  1. Supplementary angles total 180°.
  2. 180° - 83° = 97°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 97°

Worked Example 9

Problem: Find the supplementary angle to 95°.

  1. Supplementary angles total 180°.
  2. 180° - 95° = 85°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 85°

Worked Example 10

Problem: Find the supplementary angle to 120°.

  1. Supplementary angles total 180°.
  2. 180° - 120° = 60°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 60°

Practice exercise

Create one new Grade 8 problem involving Real-life square-root problems. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 7 Review Questions and Answers

Q1. What is important to remember about Square numbers to 144?

Answer: Square numbers to 144 is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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 Perfect squares?

Answer: Perfect squares is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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.

Q3. What is important to remember about Square roots?

Answer: A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Square roots.

Q4. What is important to remember about Square-root notation?

Answer: Square-root notation is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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 Memorizing squares from 1² to 12²?

Answer: Memorizing squares from 1² to 12² is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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 Finding exact square roots?

Answer: A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Finding exact square roots.

Q7. What is important to remember about Estimating non-perfect square roots?

Answer: A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Estimating non-perfect square roots.

Q8. What is important to remember about Square roots on a number line?

Answer: A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Square roots on a number line.

Q9. What is important to remember about Comparing square roots?

Answer: A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Comparing square roots.

Q10. What is important to remember about Area and square roots?

Answer: A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Area and square roots.

Q11. What is important to remember about Calculator use?

Answer: Calculator use is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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.

Q12. What is important to remember about Real-life square-root problems?

Answer: Real-life square-root problems is an important Grade 8 concept in Squares, Perfect Squares, and Square Roots. 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.

Q13. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q14. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q15. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q16. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q17. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q18. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q19. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q20. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q21. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q22. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q23. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q24. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q25. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q26. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q27. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q28. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q29. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.

Q30. Why should data sources be checked?

Answer: Reliable sources and fair collection methods make conclusions more trustworthy.