1.2: Exponent Rules | Algebra 1 Honors

How it works

Multiplying powers with the same base adds exponents; dividing subtracts them. A power of a power multiplies exponents. A negative exponent represents a reciprocal, so the base must be nonzero. These rules do not combine unlike bases by adding their exponents.

a^ma^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n}\ (a\ne0)

Worked examples

Example 1

Simplify $\displaystyle (x^3)^2x^4$ .

First apply the power rule: $\displaystyle (x^3)^2=x^6$ . Then $\displaystyle x^6x^4=x^{10}$ .

Example 2

Write $\displaystyle 2^{-3}$ as a fraction.

A negative exponent gives a reciprocal: $\displaystyle 2^{-3}=1/2^3=1/8$ .

Problems

Problem 1.2.1

Simplify $\displaystyle \frac{x^7}{x^3}$ and state a restriction.

Show worked solution

For $\displaystyle x\ne0$ , subtract exponents: $\displaystyle x^{7-3}=x^4$ . The original denominator still excludes $\displaystyle 0$ .

Problem 1.2.2

Simplify $\displaystyle (3a^2b)^2$ .

Show worked solution

Square every factor: $\displaystyle 3^2a^4b^2=9a^4b^2$ .

Problem 1.2.3

Write $\displaystyle 5x^{-2}$ using positive exponents.

Show worked solution

$\displaystyle 5x^{-2}=5/x^2$ , with $\displaystyle x\ne0$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.2.1

Problem 1.2.2

Problem 1.2.3

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