GMAT Quantitative: Exponents and Roots 1 — Questions and Answers

Question 1: Simplify: 2³ × 2⁵

2⁷
2⁸ (Correct answer)
4⁸
2¹⁵

Correct answer: 2⁸

When multiplying powers with the same base, add the exponents: 2³ × 2⁵ = 2^(3+5) = 2⁸.

Product rule for exponents: aᵐ × aⁿ = a^(m+n). Here: 2³ × 2⁵ = 2^(3+5) = 2⁸ = 256.

Question 2: Simplify: (3²)⁴

3⁶
3⁸ (Correct answer)
9⁸
3⁴

Correct answer: 3⁸

Power of a power rule: (aᵐ)ⁿ = a^(m×n). (3²)⁴ = 3^(2×4) = 3⁸.

(3²)⁴ = 3^(2 × 4) = 3⁸ = 6,561.

Question 3: What is the value of 5⁰?

0
1 (Correct answer)
5
Undefined

Correct answer: 1

Any nonzero number raised to the power 0 equals 1. 5⁰ = 1.

By definition, for any a ≠ 0: a⁰ = 1. This follows from the division rule: aⁿ ÷ aⁿ = a^(n−n) = a⁰ = 1.

Question 4: What is the value of 2⁻³?

−8
−6
1/8 (Correct answer)
1/6

Correct answer: 1/8

Negative exponent rule: a⁻ⁿ = 1/aⁿ. 2⁻³ = 1/2³ = 1/8.

2⁻³ = 1/(2³) = 1/8 = 0.125.

Question 5: Simplify: 6⁸ ÷ 6³

6⁵ (Correct answer)
6¹¹
1⁵
6²⁴

Correct answer: 6⁵

Division rule: aᵐ ÷ aⁿ = a^(m−n). 6⁸ ÷ 6³ = 6^(8−3) = 6⁵.

6⁸ ÷ 6³ = 6^(8−3) = 6⁵ = 7,776.

Question 6: What is √144?

11
12 (Correct answer)
13
14

Correct answer: 12

12² = 144, so √144 = 12.

√144 = 12 because 12 × 12 = 144. Perfect square recognition is essential for GMAT quant.

Question 7: Simplify: √(50)

5√2 (Correct answer)
10√5
5√10
25√2

Correct answer: 5√2

√50 = √(25 × 2) = √25 × √2 = 5√2.

50 = 25 × 2. √50 = √25 × √2 = 5√2 ≈ 7.07.

Question 8: What is 27^(1/3)?

3 (Correct answer)
9
6
4

Correct answer: 3

27^(1/3) is the cube root of 27. Since 3³ = 27, the answer is 3.

27^(1/3) = ∛27 = 3, since 3³ = 27. Fractional exponents: a^(1/n) = ⁿ√a.

Question 9: Simplify: (2x³)(4x²)

8x⁵ (Correct answer)
8x⁶
6x⁵
8x

Correct answer: 8x⁵

Multiply coefficients: 2 × 4 = 8. Multiply powers: x³ × x² = x^(3+2) = x⁵. Result: 8x⁵.

(2x³)(4x²) = (2 × 4)(x^(3+2)) = 8x⁵.

Question 10: If 2^x = 32, what is x?

4
5 (Correct answer)
6
3

Correct answer: 5

32 = 2⁵, so x = 5.

2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32. Therefore x = 5.

Question 11: Simplify: (x⁴)/(x⁷)

x³
x⁻³
1/x³ (Correct answer)
x¹¹

Correct answer: 1/x³

x⁴/x⁷ = x^(4−7) = x⁻³ = 1/x³.

x⁴/x⁷ = x^(4−7) = x⁻³ = 1/x³. Answers B and C are equivalent, but 1/x³ is the simplified positive form.

Question 12: What is the value of (1/2)⁻²?

1/4
4 (Correct answer)
−4
2

Correct answer: 4

(1/2)⁻² = (2/1)² = 2² = 4. A negative exponent flips the fraction.

(1/2)⁻² = (2)² = 4. General rule: (a/b)⁻ⁿ = (b/a)ⁿ.

Question 13: Which is greater: 2¹⁰ or 10²?

2¹⁰ (Correct answer)
10²
They are equal
Cannot determine

Correct answer: 2¹⁰

2¹⁰ = 1,024. 10² = 100. 1,024 > 100, so 2¹⁰ is greater.

2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024. 10² = 100. 1,024 >> 100.

Question 14: Simplify: √(x⁶)

x²
x³ (Correct answer)
x⁴
x^(3/2)

Correct answer: x³

√(x⁶) = x^(6/2) = x³ (assuming x ≥ 0).

√(x⁶) = (x⁶)^(1/2) = x^(6×1/2) = x³.

Question 15: If 3^(2x) = 81, what is x?

1
2 (Correct answer)
3
4

Correct answer: 2

81 = 3⁴. So 3^(2x) = 3⁴ → 2x = 4 → x = 2.

3⁴ = 81. Setting 3^(2x) = 3⁴: 2x = 4, x = 2.

Question 16: Simplify: (2³ × 3²) / (2 × 3)

6
12 (Correct answer)
18
24

Correct answer: 12

(2³/2) × (3²/3) = 2² × 3¹ = 4 × 3 = 12.

(2³ × 3²) / (2¹ × 3¹) = 2^(3−1) × 3^(2−1) = 2² × 3 = 4 × 3 = 12.

Question 17: What is 4^(3/2)?

6
8 (Correct answer)
12
16

Correct answer: 8

4^(3/2) = (4^(1/2))³ = (√4)³ = 2³ = 8.

4^(3/2) = (√4)³ = 2³ = 8. Alternatively: 4^(3/2) = (4³)^(1/2) = 64^(1/2) = 8.

Question 18: Simplify: (√3 + √3)²

6
12 (Correct answer)
18
4√3

Correct answer: 12

√3 + √3 = 2√3. (2√3)² = 4 × 3 = 12.

√3 + √3 = 2√3. (2√3)² = 2² × (√3)² = 4 × 3 = 12.

Question 19: If x² = 49 and x > 0, what is x³?

49
343 (Correct answer)
2,401
7

Correct answer: 343

x² = 49 → x = 7 (since x > 0). x³ = 7³ = 343.

x² = 49 → x = 7. x³ = 7 × 7 × 7 = 343.

Question 20: Which expression equals √(a²b⁴) where a, b > 0?

ab² (Correct answer)
a²b²
ab⁴
a²b⁴

Correct answer: ab²

√(a²b⁴) = √(a²) × √(b⁴) = a × b² = ab².

√(a²b⁴) = (a²b⁴)^(1/2) = a^(2/2) × b^(4/2) = a¹ × b² = ab².

Question 21: Simplify: 5^x × 5^(−x)

1 (Correct answer)
5
0
5^(x²)

Correct answer: 1

5^x × 5^(−x) = 5^(x+(−x)) = 5⁰ = 1.

5^x × 5^(−x) = 5^(x−x) = 5⁰ = 1.

Question 22: The expression ∛(8x³) equals:

2x (Correct answer)
4x
2x³
8x

Correct answer: 2x

∛(8x³) = ∛8 × ∛(x³) = 2 × x = 2x.

∛8 = 2 (since 2³ = 8). ∛(x³) = x. Therefore ∛(8x³) = 2x.

Question 23: If a = 2 and b = 3, what is a^b + b^a?

13
17
16 (Correct answer)
11

Correct answer: 16

2³ + 3² = 8 + 9 = 17. Answer is B (17).

a^b = 2³ = 8. b^a = 3² = 9. Sum = 8 + 9 = 17.

Question 24: If a = 2 and b = 3, what is a^b + b^a?

13
17 (Correct answer)
16
25

Correct answer: 17

2³ + 3² = 8 + 9 = 17.

a^b = 2³ = 8. b^a = 3² = 9. Total = 17.

Question 25: Which of the following is NOT equivalent to 8?

2³
4^(3/2)
(-2)³ (Correct answer)
2 × 4

Correct answer: (-2)³

(-2)³ = −8, not 8. All others equal 8: 2³=8, 4^(3/2)=8, 2×4=8.

2³ = 8. ✓ 4^(3/2) = (√4)³ = 2³ = 8. ✓ 2 × 4 = 8. ✓ (−2)³ = −8. ✗ Answer: C.

Question 26: Simplify: (x²y³)² / (x³y)

xy⁵ (Correct answer)
x²y⁵
xy⁴
x⁴y⁶

Correct answer: xy⁵

Numerator: (x²y³)² = x⁴y⁶. Denominator: x³y¹. Result: x^(4−3) × y^(6−1) = x¹y⁵ = xy⁵.

(x²y³)² = x⁴y⁶. Dividing: x⁴y⁶ / x³y = x^(4−3) × y^(6−1) = xy⁵.

Question 27: What is the value of 100^(1/2) × 8^(1/3)?

20 (Correct answer)
40
30
16

Correct answer: 20

100^(1/2) = √100 = 10. 8^(1/3) = ∛8 = 2. 10 × 2 = 20.

100^(1/2) = 10. 8^(1/3) = 2. Product = 10 × 2 = 20.

Question 28: If 4^x = 16^3, what is x?

4
5
6 (Correct answer)
8

Correct answer: 6

16³ = (4²)³ = 4⁶. So 4^x = 4⁶ → x = 6.

16 = 4². So 16³ = (4²)³ = 4^(2×3) = 4⁶. Therefore 4^x = 4⁶ → x = 6.

Question 29: Simplify: √(75) − √(48)

√3 (Correct answer)
3√3
2√3
5√3

Correct answer: √3

√75 = √(25×3) = 5√3. √48 = √(16×3) = 4√3. 5√3 − 4√3 = √3.

√75 = 5√3. √48 = 4√3. Difference = 5√3 − 4√3 = √3.

Question 30: What is (-1)^100?

−1
1 (Correct answer)
0
100

Correct answer: 1

(−1)^n = 1 when n is even. 100 is even, so (−1)^100 = 1.

Pattern: (−1)^even = +1, (−1)^odd = −1. 100 is even, so (−1)^100 = 1.

Question 31: The number of digits in 2^10 is:

3 (Correct answer)
4
5
2

Correct answer: 3

2^10 = 1,024. This has 4 digits. Answer B.

2^10 = 1,024. Count the digits: 1, 0, 2, 4 → 4 digits.

Question 32: How many digits does 2^10 have?

3
4 (Correct answer)
5
2

Correct answer: 4

2^10 = 1,024, which has 4 digits.

2^10 = 1,024. The number 1,024 contains 4 digits: 1, 0, 2, and 4.

Question 33: If 5^(x+1) = 125, what is x?

1
2 (Correct answer)
3
4

Correct answer: 2

125 = 5³. So 5^(x+1) = 5³ → x+1 = 3 → x = 2.

5³ = 125. Setting 5^(x+1) = 5³: x+1 = 3, so x = 2.

Question 34: Which is larger: 3⁴ or 4³?

3⁴ (Correct answer)
4³
They are equal
Cannot determine

Correct answer: 3⁴

3⁴ = 81. 4³ = 64. 81 > 64, so 3⁴ is larger.

3⁴ = 3 × 3 × 3 × 3 = 81. 4³ = 4 × 4 × 4 = 64. 81 > 64, so 3⁴ > 4³.

Question 35: Simplify: (√5)⁴

5
10
25 (Correct answer)
√5

Correct answer: 25

(√5)⁴ = (5^(1/2))⁴ = 5^(4/2) = 5² = 25.

(√5)⁴ = (5^(1/2))⁴ = 5^(1/2 × 4) = 5² = 25.

Question 36: If n is a positive integer, which of the following is always even?

n²
2n (Correct answer)
n³
n+1

Correct answer: 2n

2n is always even regardless of whether n is odd or even, since it has 2 as a factor.

2n = 2 × n, which always has 2 as a factor and is therefore always even. n² and n³ are even only when n is even. n+1 is even only when n is odd.

GMAT Quantitative: Exponents and Roots 1

Simplify: 2³ × 2⁵

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