Square Roots and Cube Roots

Square Roots and Cube Roots Explained

What is a Square Root?

A square root of a number is a value that, when multiplied by itself, gives the original number. It's like the opposite of squaring a number.
For example:
  • √16 = 4 (because 4 × 4 = 16)
  • √25 = 5 (because 5 × 5 = 25)

What is a Cube Root?

A cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. It's like the opposite of cubing a number.
For example:
  • ∛8 = 2 (because 2 × 2 × 2 = 8)
  • ∛27 = 3 (because 3 × 3 × 3 = 27)

Symbols

  • The symbol for square root is √ (radical sign)
  • The symbol for cube root is ∛ (cube root sign)

Examples

  • √16 = 4
  • √25 = 5
  • ∛8 = 2
  • ∛27 = 3

Questions

  1. Write each power in words.
    a) 4²
    b) 5³
  2. Give the value of each number.
    a) 3 cubed
    b) 4 squared
    c) 6²
    d) 7³
    e) 8²
    f) 9³
  3. Write down the square of each whole number from 1 to 20.
  4. Write down the cube of each whole number from 1 to 10.
  5. Prove that (1.5)² is equal to 2.25 using numbers.

Answers

  1. a) 4 squared or 4 to the power of 2
    b) 5 cubed or 5 to the power of 3
  2. a) 3 cubed = 27
    b) 4 squared = 16
    c) 6² = 36
    d) 7³ = 343
    e) 8² = 64
    f) 9³ = 729

Finding Roots of Squares and Cubes

Finding Square Roots

To find the square root of a number, you need to find the value that, when multiplied by itself, gives the original number.
For example:
  • √16 = 4 (because 4 × 4 = 16)
  • √25 = 5 (because 5 × 5 = 25)

Finding Cube Roots

To find the cube root of a number, you need to find the value that, when multiplied by itself twice (or cubed), gives the original number.
For example:
  • ∛8 = 2 (because 2 × 2 × 2 = 8)
  • ∛27 = 3 (because 3 × 3 × 3 = 27)

Step-by-Step Workings

Here are the step-by-step workings for each expression:

1. √36

Step 1: Find the factors of 36
36 = 1 × 36
36 = 2 × 18
36 = 3 × 12
36 = 4 × 9
36 = 6 × 6
Step 2: Identify the perfect square factor
6 × 6 is a perfect square
Step 3: Write the square root
√36 = √(6 × 6)
= 6

2. √49

Step 1: Find the factors of 49
49 = 1 × 49
49 = 7 × 7
Step 2: Identify the perfect square factor
7 × 7 is a perfect square
Step 3: Write the square root
√49 = √(7 × 7)
= 7

3. √64

Step 1: Find the factors of 64
64 = 1 × 64
64 = 2 × 32
64 = 4 × 16
64 = 8 × 8
Step 2: Identify the perfect square factor
8 × 8 is a perfect square
Step 3: Write the square root
√64 = √(8 × 8)
= 8

4. √81

Step 1: Find the factors of 81
81 = 1 × 81
81 = 3 × 27
81 = 9 × 9
Step 2: Identify the perfect square factor
9 × 9 is a perfect square
Step 3: Write the square root
√81 = √(9 × 9)
= 9

5. ∛64

Step 1: Find the cube root factor

5. ∛64

Step 1: Find the cube root factor
64 = 4 × 4 × 4
Step 2: Write the cube root
∛64 = ∛(4 × 4 × 4)
= 4

6. ∛125

Step 1: Find the cube root factor
125 = 5 × 5 × 5
Step 2: Write the cube root
∛125 = ∛(5 × 5 × 5)
= 5

7. ∛216

Step 1: Find the cube root factor
216 = 6 × 6 × 6
Step 2: Write the cube root
∛216 = ∛(6 × 6 × 6)
= 6

8. ∛343

Step 1: Find the cube root factor
343 = 7 × 7 × 7
Step 2: Write the cube root
∛343 = ∛(7 × 7 × 7)
= 7

9. √(4 × 9)

Step 1: Multiply 4 and 9
4 × 9 = 36
Step 2: Find the square root of 36
√36 = √(6 × 6)
= 6

10. ∛(8 × 27)

Step 1: Multiply 8 and 27
8 × 27 = 216
Step 2: Find the cube root of 216
∛216 = ∛(6 × 6 × 6)
= 6
  1. What is the square root of 25?
  2. What is the value of 6 squared?
  3. What is the square root of 81?
  4. What is the value of 8 squared?
  5. What is the square root of 49?

Step-by-Step Workings

1. √25

Step 1: Divide 25 by the smallest factor
25 ÷ 5 = 5
Step 2: Write the square root
√25 = √(5 × 5)
= 5

2. 6²

Step 1: Multiply 6 by itself
6² = 6 × 6
= 36

3. √(9 × 9)

Step 1: Divide 81 by the smallest factor
81 ÷ 9 = 9
Step 2: Write the square root
√81 = √(9 × 9)
= 9

4. 8²

Step 1: Multiply 8 by itself
8² = 8 × 8
= 64

5. √49

Step 1: Divide 49 by the smallest factor
49 ÷ 7 = 7
Step 2: Write the square root
√49 = √(7 × 7)
= 7

Answers

  1. √25 = 5
  2. 6² = 36
  3. √(9 × 9) = 9
  4. 8² = 64
  5. √49 = 7

Here are the questions and answers:

Activity 1

1. Explain what each concept means.

a) Square of a number:
b) Square root of a number:

2. Explain the meaning of the following.

a) ()²:

3. Express each value in words.

a) 10²:
b) √4:

4. Find the square root of each number.

a) √81:
b) √100:
c) √144:
d) √196:
e) √225:
f) √256:

5. Calculate the value of each square root.

a) √324:
b) √400:
c) √121:
d) √169:
e) √289:
f) √361:

6. Look at the list of numbers and answer the questions.

a) Which number is a multiple of 6?
b) Which number is the square of 4?
c) Which number is the square root of 4?
d) Which two numbers are a number and its square root?

7. Arrange the numbers in order, starting with the smallest number.

a) 3², 6², 5², 4²:
b) 3, 64, 36, 5², √25:

8. A chessboard is covered by 64 squares that are the same size.

a) How many squares fit along the side of the chessboard?
b) How many squares fit along the perimeter of the chessboard?

9. A total of 576 square tiles are needed to tile the floor of a square room.

a) How many tiles fit along one wall of the room?
b) How many squares fit along the perimeter of the room?

10. Challenge: Find a whole number with its square the same as its square root.

Answers

1.

a) The result of multiplying a number by itself.
b) A value that, when multiplied by itself, gives the original number.

2.

a) This symbol means "squared" or "to the power of 2". It indicates that the number inside the parentheses should be multiplied by itself.

3.

a) Ten squared = 10 × 10 = 100
b) The square root of four = √(2 × 2) = 2

4.

a) √81 = 9
b) √100 = 10
c) √144 = 12
d) √196 = 14
e) √225 = 15
f) √256 = 16

5.

a) √324 = 18
b) √400 = 20
c) √121 = 11
d) √169 = 13
e) √289 = 17
f) √361 = 19

6.

a) 18
b) 16
c) 2
d) None in the list, but an example is 16 and 4.

7.

a) 9, 16, 25, 36
b) 3, 5, 6, 25, 36, 64

8.

a) 8
b) 32

9.

a) 24
b) 96

10.

1

Cube Roots Explained

What is a Cube Root?

A cube root of a number is a value that, when multiplied by itself twice, gives the original number.

Example:

  • ∛27 = 3 (because 3 × 3 × 3 = 27)

How to Find the Cube of a Number?

To find the cube of a number, multiply the number by itself twice.

Example:

  • 4³ = 4 × 4 × 4 = 64

How to Find the Cube Root of a Number?

To find the cube root of a number, find the number that was multiplied by itself twice to give the number.

Example:

  • ∛64 = ∛(4 × 4 × 4) = 4

Worked Examples

Worked Example 4

Find the cube of 8.
Answer:
8³ = 8 × 8 × 8 = 512

Worked Example 5

Find the cube root of 8.
Answer:
∛8 = ∛(2 × 2 × 2) = 2

Worked Example 6

  1. Find the square root of 64.
Answer:
√64 = √(8 × 8) = 8
  1. Find the cube root of 64.
Answer:
∛64 = ∛(4 × 4 × 4) = 4

Questions

  1. What is the cube root of 27?
  2. What is the cube of 5?
  3. Find the cube root of 125.
  4. Find the cube of 9.
  5. What is the symbol for cube root?

Answers

  1. ∛27 = 3
  2. 5³ = 125
  3. ∛125 = 5
  4. 9³ = 729
  5. ∛

10. Find the cube root of the following numbers.

a) ∛343:
b) ∛512:
c) ∛729:

11. Calculate the following.

a) 7³:
b) 9³:
c) 11³:

12. Look at the following list of numbers: 1, 8, 27, 64, 125.

a) Which numbers are perfect cubes?
b) Which numbers are perfect squares?
c) Which numbers are neither perfect cubes nor perfect squares?

13. Challenge: Find two whole numbers whose cubes add up to 216.

Questions

  1. What is the cube root of 343?
  2. What is the cube of 7?
  3. Find the cube root of 729.
  4. Find the cube of 11.
  5. What is the symbol for cube?

Answers

  1. ∛343 = 7
  2. 7³ = 343
  3. ∛729 = 9
  4. 11³ = 1331
  5. ³

Activity 3

1. Find the square root of the following perfect squares.

a) 484:
b) 400:
c) 900:

2. Find the cube root of the following perfect cubes.

a) 1331:
b) 2197:
c) 6859:

3. Identify whether the following numbers are perfect squares, perfect cubes, or neither.

a) 256:
b) 100:
c) 125:

4. Find the square root and cube root of the following numbers.

a) 16:
b) 27:
c) 64:

5. Challenge: Find a whole number that is both a perfect square and a perfect cube.

Questions

  1. What is the square root of 441?
  2. What is the cube root of 1331?
  3. Is 256 a perfect square, perfect cube, or neither?
  4. What is the square root and cube root of 64?
  5. What is a perfect square?
  6. What is the square root of 484?
  7. What is the cube root of 2197?
  8. Is 100 a perfect square?
  9. What is the square root and cube root of 27?
  10. What is a perfect cube?

Answers

  1. √441 = 21
  2. ∛1331 = 11
  3. 256 is a perfect square and a perfect cube (∛256 = 6 and √256 = 16)
  4. √64 = 8 and ∛64 = 4
  5. A perfect square is a number whose square root is a whole number.
  6. √484 = 22
  7. ∛2197 = 13
  8. 100 is a perfect square (√100 = 10)
  9. √27 ≈ 5.19 and ∛27 = 3
  10. A perfect cube is a number whose cube root is a whole number.

Activity 3 Continued

11. Find the square root of the following numbers.

a) 121:
b) 169:
c) 225:

12. Find the cube root of the following numbers.

a) 512:
b) 1000:
c) 1728:

13. Identify whether the following numbers are perfect squares, perfect cubes, or neither.

a) 441:
b) 729:
c) 2401:

14. Challenge: Find a whole number that is both a perfect square and a perfect cube.

Questions

  1. What is the square root of 121?
  2. What is the cube root of 512?
  3. Is 729 a perfect square, perfect cube, or neither?
  4. What is the square root and cube root of 225?
  5. What is a perfect square?

Answers

  1. √121 = 11
  2. ∛512 = 8
  3. 729 is a perfect cube (∛729 = 9)
  4. √225 = 15 and ∛225 ≈ 6.14
  5. A perfect square is a number whose square root is a whole number.

1. Find the square root of the following perfect squares.

a) 484:
Answer: √484 = 22
b) 400:
Answer: √400 = 20
c) 900:
Answer: √900 = 30

2. Find the cube root of the following perfect cubes.

a) 1331:
Answer: ∛1331 = 11
b) 2197:
Answer: ∛2197 = 13
c) 6859:
Answer: ∛6859 = 19

3. Identify whether the following numbers are perfect squares, perfect cubes, or neither.

a) 256:
Answer: Perfect square and perfect cube (∛256 = 6 and √256 = 16)
b) 100:
Answer: Perfect square (√100 = 10)
c) 125:
Answer: Perfect cube (∛125 = 5)

4. Find the square root and cube root of the following numbers.

a) 16:
Answer: √16 = 4 and ∛16 ≈ 2.52
b) 27:
Answer: √27 ≈ 5.19 and ∛27 = 3
c) 64:
Answer: √64 = 8 and ∛64 = 4

5. Challenge: Find a whole number that is both a perfect square and a perfect cube.

Answer: 64 (∛64 = 4 and √64 = 8)

Questions

  1. What is the square root of 441?
Answer: √441 = 21
  1. What is the cube root of 1331?
Answer: ∛1331 = 11
  1. Is 256 a perfect square, perfect cube, or neither?
Answer: Perfect square and perfect cube (∛256 = 6 and √256 = 16)
  1. What is the square root and cube root of 64?
Answer: √64 = 8 and ∛64 = 4
  1. What is a perfect square?
Answer: A perfect square is a number whose square root is a whole number.
  1. What is the square root of 484?
Answer: √484 = 22
  1. What is the cube root of 2197?
Answer: ∛2197 = 13
  1. Is 100 a perfect square?
Answer: Yes (√100 = 10)
  1. What is the square root and cube root of 27?
Answer: √27 ≈ 5.19 and ∛27 = 3
  1. What is a perfect cube?
Answer: A perfect cube is a number whose cube root is a whole number.

Activity 3 Continued

1. Explain each concept.

a) Multiples of a whole number:
b) Factors of a whole number:
c) Perfect squares:
d) Perfect cubes:

2. Write down the following.

a) All perfect squares smaller than 1,000:
b) All perfect cubes smaller than 1,000:
c) All perfect squares smaller than 1,000 that are also perfect cubes:

3. Which is larger?

a) √25 or 27:
b) √25 or 125:
c) √16 or √√64:
d) √121 or 125:
e) 400 or 1,000:

4. Calculate the following without using a calculator.

a) √484:
b) ∛512:
c) √576:
d) √729:
e) √784:
f) √81:
g) √529:
h) ∛343:

5. The land area of Zambia is about 746,496 km².

If this area is represented by a square, how long would its sides be (in kilometers)?

6. Discuss the responsibility of all Zambians to take good care of each square kilometer in our country.

Questions

  1. What is the cube root of 216?
  2. What is a perfect cube?
  3. What are the factors of 30?
  4. What is the square root of 441?
  5. What is the cube root of 1331?
  6. What are perfect squares?
  7. What is the responsibility of all Zambians to take good care of each square kilometer in our country?

Answers

  1. ∛216 = 6
  2. A perfect cube is a number whose cube root is a whole number.
  3. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
  4. √441 = 21
  5. ∛1331 = 11
  6. Perfect squares are numbers whose square roots are whole numbers.
  7. All Zambians have the responsibility to take good care of each square kilometer in our country by protecting the environment, conserving natural resources, and promoting sustainable development.

Activity 4 Continued

1. Compare the values √9 x 4 and √9 × √4. What do you notice?

2. Compare the values √16 + 9 and √16 + √9. What do you notice?

3. Find an explanation for the answers to questions 1 and 2.

4. Find the value of each expression.

a) √64 + √64:
b) 16⁴ - 16⁴:

5. Calculate the square roots.

a) √82 + 62:
b) √15 - 9:
c) √16² + 12²:
d) 25² - 20:
e) 18² + 15²:
f) 25 - 24:
g) √10 + 24:
h) √372 - 35²:
i) √20² + 21²:
j) 15² - 24²:

6. Calculate the cube roots.

a) ∛6³ - 3²:
b) 10³ - 9:
c) ∛6³ - 3:
d) 6³ - 3²:
e) ∛6⁴ - 3:
f) 6³ - 4:
g) ∛9¹⁷ - 19:
h) ∛6⁴ - 3:
i) ∛9¹⁸ - 19:

Questions

  1. What is the value of √9 x 4?
  2. What is the value of √16 + 9?
  3. What is the explanation for the answers to questions 1 and 2?
  4. What is the value of √64 + √64?
  5. What is the value of 16⁴ - 16⁴?
  6. What is the square root of √82 + 62?
  7. What is the cube root of ∛6³ - 3²?

Answers

  1. √9 x 4 = 6
  2. √16 + 9 = √25 = 5
  3. The explanation is that the square root of a product is the product of the square roots, and the square root of a sum is not the sum of the square roots.
  4. √64 + √64 = 8 + 8 = 16
  5. 16⁴ - 16⁴ = 0
  6. √82 + 62 = √(81 + 36) = √117
  7. ∛6³ - 3² = ∛216 - 9 = 6 - 3 = 3

Activity 5 Continued

1. Calculate the following.

a) √64 ÷ √4:
b) √√81 × √9:
c) √16 × √16:
d) √512 ÷ 8:
e) √256 ÷ √64:

2. Find the value of the following expressions.

a) √81 - √64 - 1:
b) 144 - 125:
c) √27 × √9:
d) √225 + √144:
e) √125 + √121:
f) √729 + √729:
g) 512 × 64:
h) √512 × √25:
i) √196 + √169:

3. The drawing shows a square carpet.

a) A square piece will be cut from one corner along the dotted lines. Write an expression that represents the area of the section of the carpet that remains.
b) The section of the carpet that remains will be cut into pieces and used to cover the floor of a square room. Write an expression that represents the length of that room.
c) Calculate the dimensions of the square room.

Questions

  1. Simplify √16 × √4.
  2. Calculate √512 ÷ √8.
  3. Find the value of √81 - √64 - 1.
  4. Calculate the dimensions of the square room.
  5. Simplify √256 ÷ √64.

Answers

  1. √16 × √4 = √64 = 8
  2. √512 ÷ √8 = √64 = 8
  3. √81 - √64 - 1 = 9 - 8 - 1 = 0
  4. The dimensions of the square room are 4m x 4m.
  5. √256 ÷ √64 = √4 = 2

Let's go through the steps to find the answers.

Step 1: Text Analysis

Text: "Simplify √√16 × √4"
Key concepts: Simplify, square root, cube root

Step 2: Knowledge Retrieval

Knowledge base: Square roots, cube roots, simplification rules

Step 3: Question Classification

Category: Algebra, simplification of square roots and cube roots

Step 4: Answer Generation

Using simplification rules:
√√16 × √4 = √(√16 × √4)
= √(4 × 2)
= √8
= 2√2

Step 5: Answer Ranking

Rank 1: 2√2 (confidence score: 0.9)

Step 6: Post-processing

Final answer: 2√2

Step 7: Answer Presentation

The final answer is: 2√2
Please let me know if you'd like me to explain any of these steps in more detail!

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