Exponential Function Manipulation

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

Manipulating exponential expressions reduces to a few exponent rules: the product, quotient, and power rules for like bases; conversion to a common base when bases are powers of a common number; and the meanings of zero, negative, and fractional exponents. The College Board uses Topic 2.4 to test whether you can simplify a mixed-base expression to a single value or simplified form.

Same-base rules

Product

b^x \cdot b^y = b^{x + y}

Quotient

\frac{b^x}{b^y} = b^{x - y}

Power

(b^x)^y = b^{xy}

Zero, negative, fractional exponents

Zero

b^0 = 1

Negative

b^{-x} = \frac{1}{b^x}

Fractional

b^{p/q} = \sqrt[q]{b^p}

Caution: Exponent rules apply to products, quotients, and powers — not sums.

(a + b)^2 \neq a^2 + b^2

Type 1

Combine same-base expressions

Use the product, quotient, and power rules in turn. Add exponents in products, subtract in quotients, and multiply in powers.

Example 1

Simplify

\dfrac{x^5 \cdot x^3}{x^4}

. (A)

x^2

(B)

x^4

(C)

x^6

(D)

x^{12}

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Type 2

Rewrite mixed bases as a common base

Rewrite each base as a power of a common smaller base, then apply the same-base rules.

Example 2

Simplify

\dfrac{4^x}{2^{x + 1}}

. (A)

2^x

(B)

2^{x - 1}

(C)

2^{2x + 1}

(D)

2^{2 - x}

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Type 3

Fractional and negative exponents

Read fractional exponents as roots and negative exponents as reciprocals. Evaluate step by step.

Example 3

Evaluate

3^{-2} \cdot 9^{1.5}

. (A)

1

(B)

3

(C)

9

(D)

27

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