AP PreCalculus Practice Quiz: Exponential Function Manipulation

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 11 questions to check your progress.

Question 1 of 11

Which of the following expressions is equivalent to 5^{(x+4)}?

A)5^x + 5^4B)5^x * 5^4C)(5^x)^4D)20x

All Questions (11)

Which of the following expressions is equivalent to 5^{(x+4)}?

A) 5^x + 5^4

B) 5^x * 5^4

C) (5^x)^4

D) 20x

Correct Answer: B

According to the product property for exponents, b^{(m+n)} = b^{m}b^{n}. Therefore, 5^{(x+4)} can be rewritten as 5^x * 5^4.

The product property for exponents, b^{m}b^{n} = b^{(m+n)}, has a graphical interpretation for exponential functions. This property implies that a horizontal translation of the form f(x) = b^{(x+k)} is equivalent to which of the following transformations?

A) A horizontal dilation

B) A vertical translation

C) A change of the base

D) A vertical dilation

Correct Answer: D

The provided content states that the product property implies that every horizontal translation of an exponential function, f(x) = b^{(x+k)}, is equivalent to a vertical dilation. This is because b^{(x+k)} can be rewritten as b^k * b^x, where b^k is a constant that vertically scales the function b^x.

Which of the following functions represents a vertical dilation of f(x) = 3^x and is also equivalent to g(x) = 3^{(x+2)}?

A) h(x) = 9 * 3^x

B) h(x) = 2 * 3^x

C) h(x) = 3^x + 2

D) h(x) = 6 * 3^x

Correct Answer: A

Using the product property of exponents, the function g(x) = 3^{(x+2)} can be rewritten as g(x) = 3^x * 3^2. Since 3^2 = 9, the equivalent function is h(x) = 9 * 3^x. This form represents a vertical dilation of f(x) = 3^x by a factor of 9.

The function f(x) = 7^{(3x)} can be rewritten in an equivalent form using the power property for exponents. Which expression is equivalent to 7^{(3x)}?

A) 3 * 7^x

B) (7^3)^x

C) 7^3 * 7^x

D) 21^x

Correct Answer: B

According to the power property for exponents, b^{(mn)} = (b^{m})^{n}. Therefore, 7^{(3x)} can be rewritten as (7^3)^x.

The power property for exponents, (b^{m})^{n} = b^{(mn)}, implies that a horizontal dilation of an exponential function, f(x) = b^{(cx)}, is graphically equivalent to which of the following?

A) A horizontal translation

B) A vertical dilation

C) A change of the base

D) A reflection across the y-axis

Correct Answer: C

The provided content explicitly states that the power property implies that every horizontal dilation of an exponential function, f(x) = b^{(cx)}, is equivalent to a change of the base of an exponential function. This is because b^{(cx)} can be rewritten as (b^c)^x, where the new base is b^c.

The function f(x) = 4^{(2x)} is a horizontal dilation of g(x) = 4^x. Using the power property, what is an equivalent function to f(x) that shows a change in the base?

A) h(x) = 8^x

B) h(x) = 2 * 4^x

C) h(x) = 16^x

D) h(x) = 4^(x+2)

Correct Answer: C

Using the power property, f(x) = 4^{(2x)} can be rewritten as f(x) = (4^2)^x. Since 4^2 = 16, the equivalent function is h(x) = 16^x. This demonstrates how a horizontal dilation is equivalent to a change of the base from 4 to 16.

According to the negative exponent property, which of the following expressions is equivalent to 10^{-3}?

A) -1000

B) 1/10^3

C) -30

D) 10 * (-3)

Correct Answer: B

The negative exponent property states that b^{-n} = 1/b^{n}. Applying this rule, 10^{-3} is equivalent to 1/10^3.

Which expression represents the value of 64^{(1/3)}?

A) The square root of 64

B) 64 divided by 3

C) The cube root of 64

D) 1/64^3

Correct Answer: C

The content states that an exponential expression involving an exponential unit fraction, b^{(1/k)}, is the kth root of b. Therefore, 64^{(1/3)} is the 3rd root (or cube root) of 64.

The function g(x) = (1/2)^x can be rewritten using the negative exponent property. Which of the following functions is equivalent to g(x)?

A) h(x) = -2^x

B) h(x) = 2^{-x}

C) h(x) = 2^x - 1

D) h(x) = x^{-2}

Correct Answer: B

Using the negative exponent property, b^{-n} = 1/b^n. We can apply this in reverse. (1/2)^x can be written as (2^{-1})^x. Then, using the power property, (2^{-1})^x = 2^{(-1*x)} = 2^{-x}.

Consider the expression (9^x)^{(1/2)}. By applying the properties of exponents in different orders, which pair of expressions are both equivalent to the original expression?

A) 9^{(x/2)} and 3^x

B) 4.5^x and 3x

C) 9^{(x+1/2)} and 9.5^x

D) 9^{(x/2)} and 18^x

Correct Answer: A

First, applying the power property (b^m)^n = b^(mn), we get (9^x)^{(1/2)} = 9^{(x * 1/2)} = 9^{(x/2)}. Second, we can change the order and evaluate the base first. The expression can be written as (9^{(1/2)})^x. Since 9^{(1/2)} is the square root of 9, which is 3, the expression simplifies to 3^x. Both 9^{(x/2)} and 3^x are equivalent forms.

The function f(x) = 8 * 2^x can be rewritten in several equivalent forms using exponent properties. Which of the following is NOT an equivalent form of f(x)?

A) 2^3 * 2^x

B) 2^{(x+3)}

C) (2^x)^3

D) 4 * 2^{(x+1)}

Correct Answer: C

Let's analyze the options. A) 8 is 2^3, so 8 * 2^x = 2^3 * 2^x, which is equivalent. B) Using the product property, 2^3 * 2^x = 2^{(x+3)}, which is equivalent. D) We can rewrite f(x) as 4 * 2 * 2^x = 4 * 2^{(x+1)}, which is equivalent. C) (2^x)^3 simplifies to 2^{(3x)} using the power property. The function 2^{(3x)} represents a horizontal dilation, which is not equivalent to the original function f(x) = 2^{(x+3)}, which is a horizontal translation of y=2^x.