Bluebook SAT Test SAT Exponents and Radicals Practice — Questions and Answers

Question 1: Which of the following is equivalent to x^(2/3)?

∛(x²) (Correct answer)
x² ÷ x³
√(x³)
x^(3/2)

Correct answer: ∛(x²)

The rational exponent 2/3 means the cube root of x squared: x^(2/3) = ∛(x²).

Question 2: Simplify: (3x²y³)² × (2xy)³

72x⁷y⁹ (Correct answer)
72x⁷y⁸
36x⁷y⁹
54x⁷y⁹

Correct answer: 72x⁷y⁹

(3x²y³)² = 9x⁴y⁶ and (2xy)³ = 8x³y³. Product = 72x^(4+3)y^(6+3) = 72x⁷y⁹.

Question 3: What is the value of 27^(−2/3)?

1/9 (Correct answer)
9
1/27
−9

Correct answer: 1/9

27^(2/3) = (∛27)² = 3² = 9. The negative exponent gives the reciprocal: 1/9.

Question 4: If 2^x = 32, what is the value of 2^(x−3)?

2
4 (Correct answer)
8
16

Correct answer: 4

2^x = 2^5 so x = 5. Then 2^(5−3) = 2² = 4.

Question 5: Simplify: √(50x⁴y²)

5x²y√2 (Correct answer)
5xy√2
10x²y
5x²√(2y)

Correct answer: 5x²y√2

√(50x⁴y²) = √(25·2·x⁴·y²) = 5x²y√2.

Question 6: Which expression is equivalent to (x^(1/2))(x^(1/3))?

x^(1/6)
x^(5/6) (Correct answer)
x^(2/3)
x^(1/3)

Correct answer: x^(5/6)

Add exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. Result is x^(5/6).

Question 7: What is the value of (8^(1/3) + 16^(1/2)) ÷ 4^(1/2)?

2.5
3 (Correct answer)
3.5
4

Correct answer: 3

8^(1/3) = 2, 16^(1/2) = 4, 4^(1/2) = 2. (2 + 4)/2 = 3.

Question 8: Which of the following equals 3^(x+2)?

3^x + 9
9 × 3^x (Correct answer)
3^x × 2
3^(2x)

Correct answer: 9 × 3^x

3^(x+2) = 3^x × 3² = 9 × 3^x.

Question 9: Simplify ∛54 in simplest radical form.

3∛2 (Correct answer)
3∛6
2∛7
6∛2

Correct answer: 3∛2

∛54 = ∛(27 × 2) = 3∛2.

Question 10: What is the value of x if √(x + 9) − √x = 1?

16 (Correct answer)
20
25
36

Correct answer: 16

Isolate: √(x+9) = √x + 1. Square: x + 9 = x + 2√x + 1 → 8 = 2√x → √x = 4 → x = 16.

Question 11: Simplify: (x²/y)³ ÷ (x/y²)²

x⁴/y
x⁴y (Correct answer)
x⁴
x⁴y⁻¹

Correct answer: x⁴y

(x²/y)³ = x⁶/y³ and (x/y²)² = x²/y⁴. Dividing: x⁶/y³ ÷ x²/y⁴ = x⁴y.

Question 12: Which of the following equals 5^(1/2) × 20^(1/2)?

2√5
5√2
10 (Correct answer)
√100

Correct answer: 10

(5 × 20)^(1/2) = 100^(1/2) = 10.

Question 13: Determine the value of (−32)^(3/5).

−8 (Correct answer)
8
−4
4

Correct answer: −8

(−32)^(3/5) = (⁵√(−32))³ = (−2)³ = −8.

Question 14: Simplify: (a²b⁻¹)^3 × (a⁻¹b²)^2

a⁴b (Correct answer)
a⁴b⁻³
a⁴b⁻¹
a⁴

Correct answer: a⁴b

(a²b⁻¹)³ = a⁶b^(−3) and (a⁻¹b²)² = a^(−2)b⁴. Product = a^(6−2)b^(−3+4) = a⁴b.

Question 15: What is (2.5 × 10⁸) ÷ (5.0 × 10³) in scientific notation?

5.0 × 10⁴ (Correct answer)
5.0 × 10⁵
0.5 × 10⁶
2.5 × 10⁵

Correct answer: 5.0 × 10⁴

(2.5 ÷ 5.0) × 10^(8−3) = 0.5 × 10⁵ = 5.0 × 10⁴.

Question 16: If f(x) = x^(2/3) and f(k) = 4, what is k?

6
8 (Correct answer)
12
16

Correct answer: 8

k^(2/3) = 4 → k = 4^(3/2) = (√4)³ = 8.

Question 17: Which is greater: 2^30 or 3^20?

2^30
3^20 (Correct answer)
They are equal
Cannot be determined

Correct answer: 3^20

2^30 = (2³)^10 = 8^10 and 3^20 = (3²)^10 = 9^10. Since 9 > 8, we have 3^20 > 2^30.

Question 18: Simplify √(x² + 6x + 9) for x ≥ 0.

x + 3 (Correct answer)
|x + 3|
x + √3
x − 3

Correct answer: x + 3

x² + 6x + 9 = (x+3)². For x ≥ 0, x+3 > 0, so √((x+3)²) = x+3.

Question 19: What is the simplified form of (27/8)^(2/3)?

3/2
9/4 (Correct answer)
3/4
4/9

Correct answer: 9/4

(27/8)^(2/3) = (∛(27/8))² = (3/2)² = 9/4.

Question 20: Solve: 2^(3x−1) = 16

x = 5/3 (Correct answer)
x = 4/3
x = 2
x = 1

Correct answer: x = 5/3

16 = 2⁴, so 3x − 1 = 4 → x = 5/3.

Question 21: Which of the following is equivalent to x^(3/4) ÷ x^(1/4)?

x^(3/16)
x^(1/2) (Correct answer)
x^(4/4)
x³

Correct answer: x^(1/2)

x^(3/4 − 1/4) = x^(2/4) = x^(1/2).

Question 22: Which of the following equals (a⁴ × b⁶)^(1/2)?

a²b³ (Correct answer)
a²b²
a⁴b³
a³b²

Correct answer: a²b³

(a⁴b⁶)^(1/2) = a^(4/2) × b^(6/2) = a²b³.

Question 23: An exponential function is y = 3 × 2^x. What is the percent increase from x = 2 to x = 3?

50%
100% (Correct answer)
150%
200%

Correct answer: 100%

At x=2: y=12; at x=3: y=24. Percent increase = (24−12)/12 × 100 = 100%.

Question 24: What is the value of 4^(3/2) + 8^(2/3)?

10
11
12 (Correct answer)
14

Correct answer: 12

4^(3/2) = (√4)³ = 8 and 8^(2/3) = (∛8)² = 4. Sum = 8 + 4 = 12.

Question 25: Simplify: (3^n × 9^n) / 27^n

1 (Correct answer)
3^n
9^n
27^n

Correct answer: 1

9^n = 3^(2n) and 27^n = 3^(3n). Numerator = 3^n × 3^(2n) = 3^(3n). Ratio = 3^(3n)/3^(3n) = 1.

Question 26: Solve: √(2x − 3) = √(x + 7)

x = 5
x = 8
x = 10 (Correct answer)
x = 14

Correct answer: x = 10

Square both sides: 2x − 3 = x + 7 → x = 10. Verify: √17 = √17 ✓.

Question 27: What is the approximate value of 5^(1.5)?

7.07
9.66
11.18 (Correct answer)
12.50

Correct answer: 11.18

5^(1.5) = 5^(3/2) = (√5)³ ≈ (2.236)³ ≈ 11.18.

Question 28: Which expression is equivalent to (√x + √y)²?

x + y
x + y + 2xy
x + y + √(xy)
x + y + 2√(xy) (Correct answer)

Correct answer: x + y + 2√(xy)

(√x + √y)² = x + 2√(xy) + y = x + y + 2√(xy).

Bluebook SAT Test SAT Exponents and Radicals Practice

Which of the following is equivalent to x^(2/3)?

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