SAT Math: Exponentials and Radicals (Drill 1)

Question 1. The value of a car, in dollars, t years after purchase is modeled by V(t) = 25,000(0.85)^t. What does 0.85 represent in this context?

• A) The car loses $0.85 in value each year.
• B) The car retains 85% of its original value after t years for the function shown.
• C) The car retains 85% of its value from one year to the next. ✓
• D) The car's value increases by 85% each year.

Explanation: In the model V(t) = 25,000(0.85)^t, the base 0.85 is the factor by which the value is multiplied each year. Since 0.85 = 1 − 0.15, the car retains 85% of its value (or loses 15%) from one year to the next. Choice B is incorrect because the 85% retention applies year over year, not relative to the original value after t years.

Question 2. What is the value of (2³)(2⁴)?

• A) 64
• B) 128 ✓
• C) 256
• D) 4,096

Explanation: When multiplying powers with the same base, add the exponents: 2³ × 2⁴ = 2⁷ = 128. Choice A (64 = 2⁶) adds incorrectly. Choice D (4,096 = 2¹²) multiplies the exponents instead of adding them.

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

• A) 2
• B) 3
• C) 3.5
• D) 4 ✓

Explanation: Express 81 as a power of 3: 81 = 3⁴. Since 3^x = 3⁴, x = 4.

Question 4. The population of a town is modeled by P(t) = 500(1.03)^t, where t is the number of years since 2010. What was the population of the town in 2010?

• A) 500 ✓
• B) 515
• C) 530
• D) 1,030

Explanation: In 2010, t = 0. P(0) = 500(1.03)⁰ = 500(1) = 500. Any number raised to the zero power equals 1, so the initial population is simply the coefficient, 500.

Question 5. Which of the following is equivalent to √72?

• A) 4√3
• B) 3√8
• C) 6√2 ✓
• D) 8√3

Explanation: Factor 72 into a perfect square times another factor: 72 = 36 × 2. Then √72 = √36 × 2 = √36 × √2 = 6√2.