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SAT Exponents and Radicals Practice Flashcards

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

    Answer: ∛(x²)

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

  2. Simplify: (3x²y³)² × (2xy)³

    Answer: 72x⁷y⁹

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

  3. What is the value of 27^(−2/3)?

    Answer: 1/9

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

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

    Answer: 4

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

  5. Simplify: √(50x⁴y²)

    Answer: 5x²y√2

    √(50x⁴y²) = √(25·2·x⁴·y²) = 5x²y√2.

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

    Answer: x^(5/6)

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

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

    Answer: 3

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

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

    Answer: 9 × 3^x

    3^(x+2) = 3^x × 3² = 9 × 3^x.

  9. Simplify ∛54 in simplest radical form.

    Answer: 3∛2

    ∛54 = ∛(27 × 2) = 3∛2.

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

    Answer: 16

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

  11. Simplify: (x²/y)³ ÷ (x/y²)²

    Answer: x⁴y

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

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

    Answer: 10

    (5 × 20)^(1/2) = 100^(1/2) = 10.

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

    Answer: −8

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

  14. Simplify: (a²b⁻¹)^3 × (a⁻¹b²)^2

    Answer: a⁴b

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

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

    Answer: 5.0 × 10⁴

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

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

    Answer: 8

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

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

    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.

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

    Answer: x + 3

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

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

    Answer: 9/4

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

  20. Solve: 2^(3x−1) = 16

    Answer: x = 5/3

    16 = 2⁴, so 3x − 1 = 4 → x = 5/3.

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