Worksheet - Exponents, Squares, and Square Roots

DIRECTIONS: Each question has five answer choices. Select the one best answer. Do not use a calculator.

Section A - Foundational Exponents and Squares - Questions 1 to 7

1. What is the value of 5³?

1. 15
2. 25
3. 75
4. 125
5. 243

2. What is the value of 12²?

1. 24
2. 121
3. 144
4. 169
5. 196

3. Which of the following is equal to \(\sqrt{81}\)?

1. 7
2. 8
3. 9
4. 10
5. 11

4. What is the value of 2⁶?

1. 12
2. 32
3. 48
4. 64
5. 128

5. What is \(\sqrt{196}\)?

1. 12
2. 13
3. 14
4. 15
5. 16

6. Which expression is equivalent to 3⁴?

1. 3 × 4
2. 4 × 4 × 4
3. 3 × 3 × 3
4. 3 × 3 × 3 × 3
5. 3 + 3 + 3 + 3

7. What is the value of 15² - 13²?

1. 2
2. 4
3. 28
4. 56
5. 225

Section B - Multi-Step Operations - Questions 8 to 14

8. What is the value of \(\frac{10^3}{10^2}\)?

1. 1
2. 10
3. 100
4. 1000
5. 10,000

9. If \(x^2 = 64\), what are the possible values of \(x\)?

1. 8 only
2. -8 only
3. 8 and -8
4. 32 only
5. 4 and -4

10. What is the value of \(3^2 + 4^2\)?

1. 7
2. 25
3. 49
4. 144
5. 169

11. Which of the following is equal to \(2^3 \times 2^4\)?

1. 2⁷
2. 2¹²
3. 4⁷
4. 8⁴
5. 16³

12. What is \(\sqrt{225} - \sqrt{144}\)?

1. 3
2. 9
3. 12
4. 15
5. 81

13. If \(5^n = 625\), what is the value of \(n\)?

1. 3
2. 4
3. 5
4. 25
5. 125

14. What is the value of \((2^3)^2\)?

1. 32
2. 36
3. 64
4. 128
5. 256

Section C - Advanced Application - Questions 15 to 20

15. The square of a positive number is 361. What is the number?

1. 17
2. 18
3. 19
4. 20
5. 21

16. A square garden has an area of 289 square feet. What is the length of one side of the garden?

1. 13 feet
2. 14 feet
3. 15 feet
4. 16 feet
5. 17 feet

17. If \(3^x = 81\) and \(2^y = 16\), what is the value of \(x + y\)?

1. 6
2. 7
3. 8
4. 9
5. 10

18. What is the value of \(\sqrt{16 + 9}\)?

1. 4
2. 5
3. 7
4. 13
5. 25

19. A number is squared, then 20 is added to the result, giving a final answer of 69. What was the original number?

1. 5
2. 6
3. 7
4. 8
5. 9

20. Which of the following expressions has the greatest value?

1. \(2^5\)
2. \(3^3\)
3. \(4^2 + 10\)
4. \(5^2 - 1\)
5. \(\sqrt{900}\)

Answer Key

Quick Reference

1 D 2 C 3 C 4 D 5 C 6 D 7 D 8 B 9 C 10 B

11 A 12 A 13 B 14 C 15 C 16 E 17 C 18 B 19 C 20 A

Detailed Explanations

Question 1 - Correct Answer: D

5³ means 5 × 5 × 5.
5 × 5 = 25
25 × 5 = 125

Choice A results from multiplying 5 × 3 instead of raising 5 to the third power.

Question 2 - Correct Answer: C

12² means 12 × 12.
12 × 12 = 144

Choice A results from multiplying 12 × 2 instead of squaring 12.

Question 3 - Correct Answer: C

\(\sqrt{81}\) asks what number multiplied by itself equals 81.
9 × 9 = 81
\(\sqrt{81} = 9\)

Choice B results from miscalculating the square root, possibly confusing it with \(\sqrt{64} = 8\).

Question 4 - Correct Answer: D

2⁶ means 2 × 2 × 2 × 2 × 2 × 2.
2 × 2 = 4
4 × 2 = 8
8 × 2 = 16
16 × 2 = 32
32 × 2 = 64

Choice E results from calculating 2⁷ instead of 2⁶.

Question 5 - Correct Answer: C

\(\sqrt{196}\) asks what number multiplied by itself equals 196.
14 × 14 = 196
\(\sqrt{196} = 14\)

Choice A results from calculating \(\sqrt{144}\) instead of \(\sqrt{196}\).

Question 6 - Correct Answer: D

3⁴ means 3 multiplied by itself 4 times.
3⁴ = 3 × 3 × 3 × 3

Choice A results from confusing exponentiation with multiplication, treating 3⁴ as 3 × 4.

Question 7 - Correct Answer: D

15² = 15 × 15 = 225
13² = 13 × 13 = 169
15² - 13² = 225 - 169 = 56

Choice B results from subtracting the bases first (15 - 13 = 2) and then squaring (2² = 4), which reverses the proper order of operations.

Question 8 - Correct Answer: B

10³ = 1000
10² = 100
\(\frac{10^3}{10^2} = \frac{1000}{100} = 10\)
Alternatively, using exponent laws: \(\frac{10^3}{10^2} = 10^{3-2} = 10^1 = 10\)

Choice C results from incorrectly adding the exponents instead of subtracting them.

Question 9 - Correct Answer: C

If \(x^2 = 64\), then \(x = \sqrt{64}\) or \(x = -\sqrt{64}\).
\(\sqrt{64} = 8\)
Both 8² = 64 and (-8)² = 64.
The possible values are 8 and -8.

Choice A results from forgetting that both positive and negative numbers square to give positive results.

Question 10 - Correct Answer: B

3² = 3 × 3 = 9
4² = 4 × 4 = 16
3² + 4² = 9 + 16 = 25

Choice A results from adding 3 + 4 = 7 before squaring, computing (3 + 4)² incorrectly.

Question 11 - Correct Answer: A

When multiplying powers with the same base, add the exponents.
\(2^3 \times 2^4 = 2^{3+4} = 2^7\)

Choice B results from incorrectly multiplying the exponents (3 × 4 = 12) instead of adding them.

Question 12 - Correct Answer: A

\(\sqrt{225} = 15\) because 15 × 15 = 225
\(\sqrt{144} = 12\) because 12 × 12 = 144
\(\sqrt{225} - \sqrt{144} = 15 - 12 = 3\)

Choice E results from subtracting under the radical first: \(\sqrt{225 - 144} = \sqrt{81} = 9\), which applies the operations in the wrong order.

Question 13 - Correct Answer: B

5ⁿ = 625
5¹ = 5
5² = 25
5³ = 125
5⁴ = 625
Therefore, n = 4.

Choice C results from confusing the exponent with the base, possibly thinking 5⁵ = 625.

Question 14 - Correct Answer: C

(2³)² means raising 2³ to the power of 2.
2³ = 8
8² = 64
Alternatively, using exponent laws: (2³)² = 2^3×2 = 2⁶ = 64

Choice A results from incorrectly adding the exponents: 2³⁺² = 2⁵ = 32.

Question 15 - Correct Answer: C

Let the number be \(x\).
\(x^2 = 361\)
\(x = \sqrt{361}\)
19 × 19 = 361
The number is 19.

Choice B results from miscalculating the square root, possibly computing 18 × 18 = 324 and not checking further.

Question 16 - Correct Answer: E

The area of a square is side × side = side².
side² = 289
side = \(\sqrt{289}\)
17 × 17 = 289
The length of one side is 17 feet.

Choice A results from calculating \(\sqrt{169} = 13\) instead of \(\sqrt{289}\).

Question 17 - Correct Answer: C

3^x = 81
3⁴ = 81, so x = 4.
2^y = 16
2⁴ = 16, so y = 4.
x + y = 4 + 4 = 8

Choice A results from miscalculating one of the exponents, such as thinking 3³ = 81 and getting x = 3, leading to x + y = 3 + 3 = 6.

Question 18 - Correct Answer: B

First, add inside the radical.
16 + 9 = 25
\(\sqrt{16 + 9} = \sqrt{25} = 5\)

Choice C results from incorrectly splitting the radical: \(\sqrt{16} + \sqrt{9} = 4 + 3 = 7\), which violates the properties of square roots.

Question 19 - Correct Answer: C

Let the original number be \(n\).
\(n^2 + 20 = 69\)
\(n^2 = 69 - 20\)
\(n^2 = 49\)
\(n = \sqrt{49} = 7\)

Choice C results from correctly solving the equation step by step.

Question 20 - Correct Answer: A

Calculate each expression.
Choice A: 2⁵ = 32
Choice B: 3³ = 27
Choice C: 4² + 10 = 16 + 10 = 26
Choice D: 5² - 1 = 25 - 1 = 24
Choice E: \(\sqrt{900} = 30\)
The greatest value is 32.

Choice E might appear greatest at first glance because 900 is the largest number written, but evaluating \(\sqrt{900} = 30\) shows it is less than 32.