SAT Exponents and Radicals Practice Flashcards

What is the simplified value of (16^(3/4) × 8^(−2/3)) ÷ (4^(1/2) × 2^(−1))?

Answer: 2

Convert every factor to a power of 2: 16^(3/4) = (2^4)^(3/4) = 2^3 = 8; 8^(−2/3) = (2^3)^(−2/3) = 2^(−2) = 1/4; 4^(1/2) = 2; 2^(−1) = 1/2. Numerator: 8 × (1/4) = 2. Denominator: 2 × (1/2) = 1. Final result: 2 ÷ 1 = 2.

If 4^x = 7, what is the value of 8^x ÷ 2^x?

Answer: 7

Rewrite using base 2: 8^x ÷ 2^x = 2^(3x) ÷ 2^x = 2^(2x). Notice that 2^(2x) = (2^2)^x = 4^x = 7.

For x > 0, which expression is equivalent to (x^6 ÷ √x)^(1/4)?

Answer: x^(11/8)

Rewrite √x as x^(1/2). Inside the parentheses: x^6 ÷ x^(1/2) = x^(6 − 1/2) = x^(11/2). Raising to the 1/4 power: (x^(11/2))^(1/4) = x^(11/8).

The equation 3^(2x) − 10 · 3^x + 9 = 0 has real solutions. What is the product of all those solutions?

Answer: 0

Substitute u = 3^x to obtain u² − 10u + 9 = 0, which factors as (u − 9)(u − 1) = 0. So u = 9 gives 3^x = 9 → x = 2, and u = 1 gives 3^x = 1 → x = 0. The two solutions are x = 0 and x = 2, and their product is 0 × 2 = 0.

Which of the following is equivalent to (3 + √5) ÷ (√5 − 1) in simplest radical form?

Answer: √5 + 2

Multiply numerator and denominator by the conjugate (√5 + 1). Numerator: (3 + √5)(√5 + 1) = 3√5 + 3 + 5 + √5 = 4√5 + 8. Denominator: (√5)² − 1² = 5 − 1 = 4. Result: (4√5 + 8) / 4 = √5 + 2.

If x > 0, x^a = 64, and x^b = 4, what is the value of x^(a − 3b)?

Answer: 1

Use the exponent subtraction rule: x^(a − 3b) = x^a ÷ x^(3b) = x^a ÷ (x^b)^3 = 64 ÷ (4)^3 = 64 ÷ 64 = 1.