Proving Exponent Properties

State the Domain Before Using a Rule

Exponent rules depend on the kind of exponent and the sign of the base. The following seven identities are useful, but their conditions are part of each statement.

Property	Identity	Conditions used here
Same-base product	$a^m a^n=a^{m+n}$	$a\neq0$ , $m,n\in\mathbb{Z}$
Same-base quotient	$\frac{a^m}{a^n}=a^{m-n}$	$a\neq0$ , $m,n\in\mathbb{Z}$
Power of a power	$(a^m)^n=a^{mn}$	$a\neq0$ , $m,n\in\mathbb{Z}$
Power of a product	$(ab)^m=a^m b^m$	$a,b\neq0$ , $m\in\mathbb{Z}$
Power of a quotient	$\left(\frac ab\right)^m=\frac{a^m}{b^m}$	$a,b\neq0$ , $m\in\mathbb{Z}$
Equal-denominator rational powers	$a^{m/n}a^{p/n}=a^{(m+p)/n}$	$a>0$ , $m,n,p\in\mathbb{Z}$ , $n\neq0$
General rational powers	$a^{m/n}a^{p/q}=a^{(mq+pn)/(nq)}$	$a>0$ , $m,n,p,q\in\mathbb{Z}$ , $n,q\neq0$

The integer rules can sometimes be extended to other domains, but those extensions require extra conditions. Under the conditions listed in the table, every expression is well-defined over the real numbers.

Proofs for Integer Exponents

The proofs below write each power as repeated multiplication. That form shows directly why the exponents add, subtract, or multiply. Read each proof as a short chain of equalities, and check that every step uses only the definition of a power.

Same Base Product

For positive integers $m$ and $n$ , write each power as repeated multiplication:

a^m=\underbrace{a\cdot a\cdots a}_{m\text{ factors}}

The zero case follows from $a^0=1$ . Negative integer exponents follow from $a^{-k}=\frac1{a^k}$ , which is why $a$ must be nonzero.

Same Base Quotient

Division removes $n$ factors from the numerator when $m\geq n$ :

\frac{a^m}{a^n}=\frac{\overbrace{a\cdots a}^{m\text{ factors}}}{\underbrace{a\cdots a}_{n\text{ factors}}}=a^{m-n}

If $m<n$ , the uncancelled factors stay in the denominator. The negative-exponent definition gives the same result:

\frac{a^m}{a^n}=\frac1{a^{n-m}}=a^{-(n-m)}=a^{m-n}

Power of a Power

For positive $n$ , the factor $a^m$ appears $n$ times:

\begin{aligned} (a^m)^n &=\underbrace{a^m a^m\cdots a^m}_{n\text{ factors}} \\ &=a^{m+m+\cdots+m} \\ &=a^{mn} \end{aligned}

The zero exponent covers $n=0$ , and the reciprocal covers negative $n$ .

Power of a Product and Quotient

For positive $m$ , regrouping the factors gives:

(ab)^m=\underbrace{(ab)(ab)\cdots(ab)}_{m\text{ factors}}=a^m b^m

The zero and negative cases again follow from $x^0=1$ and $x^{-k}=\frac1{x^k}$ . Requiring nonzero bases makes all needed reciprocals valid.

Proofs for Rational Exponents

For rational exponents, use a positive base so every real root is defined consistently. The proofs below extend each integer rule to fractional exponents by writing the root as a power. So every root can be rewritten with the exponent rules you already know.

Equal Denominators

Apply the same-base product rule and add the fractions:

a^{m/n}a^{p/n}=a^{m/n+p/n}

Different Denominators

First rewrite the exponents with the common denominator $nq$ :

a^{m/n}a^{p/q}=a^{m/n+p/q}

Using and Verifying Exponent Rules

The examples below apply the rules and then check the result. Each one names the rule used, so the steps can be retraced. Cover the answer, work the problem yourself, and use the named rule to find where the two solutions differ.

Determining an Unknown Exponent

The two problems below ask for an exponent that makes both sides match. Rewrite each side with the same base first, because equal bases force the exponents to be equal.

Determine $p$ in $(3^4)^2=3^p$ .
$(3^4)^2=3^{4\cdot2}=3^8$
Verify $b^4 b^5=b^9$ for $b\neq0$ .
$b^4 b^5=b^{4+5}$
Both sides have the same base and exponent, so the identity is verified.
Determine $p$ in $(3\pi)^p=27\pi^3$ .
$27\pi^3=3^3\pi^3=(3\pi)^3$

Simplifying Expressions

Each expression below combines several exponent rules. Work from the inside out, and reduce the numbers and the variables separately so a mistake in one part does not hide in the other.

Simplify $\left(\frac{2^4 3^6}{2^3 3^2}\right)^3$ .
$\left(\frac{2^4 3^6}{2^3 3^2}\right)^3=\left(2^{4-3}3^{6-2}\right)^3$
Simplify $(3u^3v^5)(9u^4v)$ .
$(3u^3v^5)(9u^4v)=27u^{3+4}v^{5+1}$
For $n\neq0$ and $r\neq0$ , simplify $\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2$ .
$\begin{aligned} &\left(\frac{n^{-1}r^4}{5n^{-6}r^{-4}}\right)^2 \\ &=\left(\frac{n^{-1-(-6)}r^{4-(-4)}}5\right)^2 \end{aligned}$