SAT Exponents & Roots – Learningscapes Academy

Example 1

If 2^(x-5) = 8 what is the value of x?

(A) 4
(B) 5
(C) 7
(D) 8

Correct Answer: (D) Use your knowledge of the exponential values of 2 to see a shortcut. Since 2^3 = 8, simply set the exponent of 2 equal to 3 to solve for x: x - 5 = 3 → x = 8 So, choice (D) is correct. Alternatively, you could work your way backwards from the answer choices. When you plug in choice (D), you will also arrive at the correct answer: 2^(x-5) = 8 Plug in 8 for x → 2^(8-5) = 8 → 2^3 = 8 Your knowledge of the exponential values of 2 will also help you quickly see the correct answer when you solve by plugging in the answer choices.

Example 2

In the table below, which of the following defines the function f(x)?

x	1	2	3	4
f(x)	1	4	9	16

(A) f(x) = x + 3
(B) f(x) = x^2
(C) f(x) = -x^2
(D) f(x) = x^3

Correct Answer: (B) Use your knowledge of the exponential values of simple numbers to recognize the pattern. 1^2 = 1 2^2 = 4 3^2 = 9 4^2 = 16 Choice (B), f(x) = x^2, therefore correctly models this relationship.

Practice Drill - Question 1

Which of these expressions is equivalent to ⁴√x³?

A) x^(4/1)
B) x^(3/4)
C) x^(4/3)
D) x^(7)

Correct Answer: B ⁴√x³: The root, 4, is on the outside = x^(3/4)

Practice Drill - Question 2

If 3x² - 27 = 0, and x > 0, what is the value of x?

A) 1
B) 2
C) 3
D) 4

Correct Answer: C 3x² - 27 = 0 → 3x² = 27 → x² = 9 → x = 3 Alternatively, you could have tried plugging in the answer choices to the original equation.

Practice Drill - Question 3

f(x) = 5^x / 2^x What is the value of f(2) in the function above?

A) 25/4
B) 5/2
C) 10/4
D) 8

Correct Answer: A f(x) = 5^x / 2^x → f(2) = 5^2 / 2^2 → f(2) = 25/4

Practice Drill - Question 4

If f(x) = √x / x what is f(9)?

A) 1/9
B) 1/3
C) 1
D) 3

Correct Answer: B f(x) = √x / x → f(9) = √9 / 9 → f(9) = 3/9 = 1/3

Practice Drill - Question 5

If x is positive, which expression is equivalent to ³√(27x⁶)?

A) 3x
B) 3x²
C) 6x²
D) 9x³

Correct Answer: B ³√(27x⁶) → Take the cube root of both 27 and x⁶ → = 3x²

Practice Drill - Question 6

Based on the values in the table above, which of the following correctly expresses the value of f(x)?

x	1	2	3	4	5
f(x)	2	5	10	17	26

A) f(x) = x + 1
B) f(x) = x²
C) f(x) = x³
D) f(x) = x² + 1

Correct Answer: D Try plugging some sample values into the answer choices to determine the answer. When you try the first ordered pair, (1, 2), choices (B) and (C) can be eliminated. When you try the next ordered pair, (2, 5), choice (A) can be eliminated. Only choice (D), f(x) = x² + 1, correctly models the function.

Practice Drill - Question 7

(xᵃ * y⁻ᵇ) / (x⁻ᵃ * yᵇ) Consider the expression above, in which x and y are both positive, and a and b are positive constants. Which of the following would be an equivalent form of the expression?

A) xᵃ / yᵇ
B) x^(a²) / y^(b²)
C) x²ᵃ / y²ᵇ
D) y / x

Correct Answer: C On a problem like this, it can be helpful to look ahead to the answers to have a sense of what the answer might look like. It appears we want a solution that has only the x values on either the numerator or denominator and only the y values on the opposite side. With that in mind, simplify the expression: (xᵃ * y⁻ᵇ) / (x⁻ᵃ * yᵇ) → (xᵃ * xᵃ) / (yᵇ * yᵇ) → x²ᵃ / y²ᵇ

Fill-In Practice - Question 8

³√125 + ³√64 = x² Given that x > 0, what is the value of x in the above equation?

Correct Answer: 3 ³√125 + ³√64 = x² → 5 + 4 = 9 = x² → 3 = x

Fill-In Practice - Question 9

If x⁻² * y⁻¹ = 1/5; what is the value of x⁴y²?

Correct Answer: 25 x⁻² * y⁻¹ = 1/5 → Flip both sides → x²y = 5 → Square both sides to make it the desired expression → x⁴y² = 25

Fill-In Practice - Question 10

If √(x/2) = 4 what is ⁵√x ?

Correct Answer: 2 √(x/2) = 4 → Square both sides → x/2 = 16 → x = 32 Take the fifth root of 32 → ⁵√32 = 2

Exponents & Roots - MCQs Practice 1

Answers will be provided at the end

Exponents and Roots