📝 Chapter 3: Exponential Functions

Assessment AS Learning — Practice Quiz

🔄 Not Graded — Unlimited Retakes

Purpose: Self-check exponent laws, rational exponents, and the exponential function family.

Score: 0 / 12

Topic 3.1 — Exponent Laws

Question 1

Simplify: \( x^7 \cdot x^{-3} \).

\( x^4 \)
\( x^{10} \)
\( x^{-21} \)
\( x^{-4} \)

Solution:

Product law: \( x^7 \cdot x^{-3} = x^{7+(-3)} = x^4 \).

Question 2

Evaluate: \( (-3)^4 \).

Solution:

\( (-3)^4 = (-3)(-3)(-3)(-3) = 81 \). Even exponent → positive result.

Question 3

Simplify: \( \dfrac{(2x^3)^2}{4x^4} \).

\( x^2 \)
\( 2x^2 \)
\( x^4 \)
\( 4x^2 \)

Solution:

\( (2x^3)^2 = 4x^6 \). Then \( \frac{4x^6}{4x^4} = x^{6-4} = x^2 \).

Topic 3.2 — Rational Exponents

Question 4

Evaluate: \( 64^{2/3} \).

Solution:

\( 64^{2/3} = (\sqrt[3]{64})^2 = 4^2 = 16 \).

Question 5

Evaluate: \( 25^{-1/2} \).

Solution:

\( 25^{-1/2} = \frac{1}{25^{1/2}} = \frac{1}{5} = 0.2 \).

Question 6

Express \( \sqrt[5]{x^3} \) using rational exponents.

\( x^{5/3} \)
\( x^{3/5} \)
\( x^{15} \)
\( x^{1/15} \)

Solution:

\( \sqrt[n]{a^m} = a^{m/n} \). So \( \sqrt[5]{x^3} = x^{3/5} \).

Topic 3.3 — Graphs of Exponentials

Question 7

The y-intercept of \( y = 5(2)^x \) is:

Solution:

At \( x = 0 \): \( y = 5(2)^0 = 5(1) = 5 \).

Question 8

Which function describes exponential decay?

\( y = 2(3)^x \)
\( y = 3(\tfrac{1}{2})^x \)
\( y = 5(1.05)^x \)
\( y = 4 \cdot 10^x \)

Solution:

Decay requires \( 0 < b < 1 \). Only \( b = \tfrac{1}{2} \) satisfies this.

Question 9

The horizontal asymptote of \( y = 3 \cdot 2^x - 4 \) is:

\( y = 0 \)
\( y = 3 \)
\( y = -4 \)
\( y = 4 \)

Solution:

The vertical shift \( c = -4 \) moves the asymptote from \( y = 0 \) to \( y = -4 \).

Topic 3.4–3.5 — Transformations & Comparing Growth

Question 10

For \( y = (\tfrac{1}{3})^x \), find \( y \) when \( x = -2 \).

Solution:

\( y = (1/3)^{-2} = 3^2 = 9 \).

Question 11

A table has consecutive y-values of 6, 18, 54, 162. The common ratio is:

Solution:

\( 18/6 = 3, 54/18 = 3, 162/54 = 3 \). Constant ratio → exponential.

Question 12

Which function eventually grows fastest as \( x \to \infty \)?

\( y = 1000x \)
\( y = x^3 \)
\( y = 2^x \)
\( y = x^{10} \)

Solution:

Exponential growth eventually beats any polynomial growth, no matter how large the polynomial degree.