,MATH101/MATH 101 Module Exam 3 | College Algebra
| Portage Learning | Q & A | 2026 Edition (PDF)
1. Which of the following is the correct definition of an exponential function?
A) A function of the form \( f(x) = x^n \), where \( n \) is a constant
B) A function of the form \( f(x) = b^x \), where \( b > 0 \) and \( b \neq 1 \)
C) A function of the form \( f(x) = \log_b(x) \), where \( b > 0 \) and \( b \neq 1 \)
D) A function of the form \( f(x) = mx + b \)
Correct Answer: A function of the form \( f(x) = b^x \), where \( b > 0 \) and \( b \neq 1 \)
Rationale: An exponential function has a variable in the exponent and a constant base, with the base
restricted to positive values not equal to 1. Power functions have a variable base and constant
exponent. Logarithmic functions are inverses of exponential functions.
2. Evaluate \( 2^5 \).
A) 10
B) 25
C) 32
D) 64
Correct Answer: 32
Rationale: \( 2^5 \) means \( 2 \times 2 \times 2 \times 2 \times 2 = 32 \).
3. Simplify \( 3^2 \cdot 3^4 \).
A) \( 3^6 \)
B) \( 3^8 \)
C) \( 9^6 \)
D) \( 3^2 \)
,Correct Answer: \( 3^6 \)
Rationale: When multiplying exponential expressions with the same base, add the exponents: \( 3^{2+4}
= 3^6 \).
4. Simplify \( \frac{5^7}{5^4} \).
A) \( 5^{11} \)
B) \( 5^3 \)
C) \( 5^{28} \)
D) \( 5^{-3} \)
Correct Answer: \( 5^3 \)
Rationale: When dividing exponential expressions with the same base, subtract the exponents: \( 5^{7-
4} = 5^3 \).
5. Simplify \( (2^3)^4 \).
A) \( 2^7 \)
B) \( 2^{12} \)
C) \( 2^{64} \)
D) \( 2^{-1} \)
Correct Answer: \( 2^{12} \)
Rationale: When raising a power to a power, multiply the exponents: \( 2^{3 \times 4} = 2^{12} \).
6. Evaluate \( 4^{-2} \).
A) \( -16 \)
, B) \( \frac{1}{16} \)
C) \( 16 \)
D) \( \frac{1}{8} \)
Correct Answer: \( \frac{1}{16} \)
Rationale: A negative exponent indicates a reciprocal: \( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \).
7. Evaluate \( \left(\frac{2}{3}\right)^3 \).
A) \( \frac{6}{9} \)
B) \( \frac{8}{27} \)
C) \( \frac{2}{27} \)
D) \( \frac{8}{9} \)
Correct Answer: \( \frac{8}{27} \)
Rationale: \( \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \).
8. Simplify \( (2x^3 y^2)^4 \).
A) \( 16x^{12} y^8 \)
B) \( 8x^{12} y^8 \)
C) \( 16x^7 y^6 \)
D) \( 8x^7 y^6 \)
Correct Answer: \( 16x^{12} y^8 \)
Rationale: Apply the exponent to each factor: \( 2^4 = 16 \), \( (x^3)^4 = x^{12} \), and \( (y^2)^4 =
y^8 \). The result is \( 16x^{12} y^8 \).
| Portage Learning | Q & A | 2026 Edition (PDF)
1. Which of the following is the correct definition of an exponential function?
A) A function of the form \( f(x) = x^n \), where \( n \) is a constant
B) A function of the form \( f(x) = b^x \), where \( b > 0 \) and \( b \neq 1 \)
C) A function of the form \( f(x) = \log_b(x) \), where \( b > 0 \) and \( b \neq 1 \)
D) A function of the form \( f(x) = mx + b \)
Correct Answer: A function of the form \( f(x) = b^x \), where \( b > 0 \) and \( b \neq 1 \)
Rationale: An exponential function has a variable in the exponent and a constant base, with the base
restricted to positive values not equal to 1. Power functions have a variable base and constant
exponent. Logarithmic functions are inverses of exponential functions.
2. Evaluate \( 2^5 \).
A) 10
B) 25
C) 32
D) 64
Correct Answer: 32
Rationale: \( 2^5 \) means \( 2 \times 2 \times 2 \times 2 \times 2 = 32 \).
3. Simplify \( 3^2 \cdot 3^4 \).
A) \( 3^6 \)
B) \( 3^8 \)
C) \( 9^6 \)
D) \( 3^2 \)
,Correct Answer: \( 3^6 \)
Rationale: When multiplying exponential expressions with the same base, add the exponents: \( 3^{2+4}
= 3^6 \).
4. Simplify \( \frac{5^7}{5^4} \).
A) \( 5^{11} \)
B) \( 5^3 \)
C) \( 5^{28} \)
D) \( 5^{-3} \)
Correct Answer: \( 5^3 \)
Rationale: When dividing exponential expressions with the same base, subtract the exponents: \( 5^{7-
4} = 5^3 \).
5. Simplify \( (2^3)^4 \).
A) \( 2^7 \)
B) \( 2^{12} \)
C) \( 2^{64} \)
D) \( 2^{-1} \)
Correct Answer: \( 2^{12} \)
Rationale: When raising a power to a power, multiply the exponents: \( 2^{3 \times 4} = 2^{12} \).
6. Evaluate \( 4^{-2} \).
A) \( -16 \)
, B) \( \frac{1}{16} \)
C) \( 16 \)
D) \( \frac{1}{8} \)
Correct Answer: \( \frac{1}{16} \)
Rationale: A negative exponent indicates a reciprocal: \( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \).
7. Evaluate \( \left(\frac{2}{3}\right)^3 \).
A) \( \frac{6}{9} \)
B) \( \frac{8}{27} \)
C) \( \frac{2}{27} \)
D) \( \frac{8}{9} \)
Correct Answer: \( \frac{8}{27} \)
Rationale: \( \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \).
8. Simplify \( (2x^3 y^2)^4 \).
A) \( 16x^{12} y^8 \)
B) \( 8x^{12} y^8 \)
C) \( 16x^7 y^6 \)
D) \( 8x^7 y^6 \)
Correct Answer: \( 16x^{12} y^8 \)
Rationale: Apply the exponent to each factor: \( 2^4 = 16 \), \( (x^3)^4 = x^{12} \), and \( (y^2)^4 =
y^8 \). The result is \( 16x^{12} y^8 \).
