SAT Math: Exponentials and Radicals (Drill 3)

Question 1. If 3^2x = 81, what is the value of x?

• A) 2 ✓
• B) 3
• C) 4
• D) 6

Explanation: 81 = 3⁴, so 3^2x = 3⁴. Therefore 2x = 4 and x = 2.

Question 2. Which expression is equivalent to (x⁴y⁶)^1/2?

• A) x²y³ ✓
• B) x²y⁴
• C) x⁸y¹²
• D) x⁴y³

Explanation: Apply the power rule: (x⁴)^1/2 = x² and (y⁶)^1/2 = y³. Result: x²y³.

Question 3. A population of bacteria triples every 5 hours. If the initial population is 200, which function models the population P after t hours?

• A) P(t) = 200(3)^t/5 ✓
• B) P(t) = 200(3)^5t
• C) P(t) = 200(5)^t/3
• D) P(t) = 600(3)^t

Explanation: Triples every 5 hours means the base is 3 and the exponent is t/5. Initial value 200 gives P(t) = 200(3)^t/5.

Question 4. Which of the following is equivalent to x^2/3 · x^1/3?

• A) x ✓
• B) x²
• C) x^1/3
• D) x^2/9

Explanation: When multiplying expressions with the same base, add the exponents: x^2/3 · x^1/3 = x^{2/3 + 1/3} = x^3/3 = x¹ = x. Choice B would require the exponents to sum to 2. Choice C (x^1/3) would require the exponents to sum to 1/3. Choice D results from multiplying the exponents (2/3 × 1/3 = 2/9) rather than adding them, a common error.

Question 5. If √5x + 1 = 6, what is the value of x?

• A) 5
• B) 7 ✓
• C) 11
• D) 35

Explanation: Square both sides: 5x + 1 = 36. Then 5x = 35, so x = 7. Check: √36 = 6. ✓