Exponent Properties

Reading Powers of Two

The table shows a repeated pattern. Moving down one row multiplies the value by $2$ because the exponent increases by one.

$2^n$	Value
$2^1$	$2$
$2^2$	$4$
$2^3$	$8$
$2^4$	$16$
$2^5$	$32$
$2^6$	$64$
$2^7$	$128$
$2^8$	$256$
$2^9$	$512$
$2^{10}$	$1024$

For example, $2^5\cdot2^3$ combines five factors of two with three more factors. The result has eight factors of two, so $2^5\cdot2^3=2^8$ .

Seven Exponent Properties

Situation	Property	Conditions used here
Multiply equal bases	$a^m a^n=a^{m+n}$	$a\neq0$ , $m,n\in\mathbb{Z}$
Divide equal bases	$\frac{a^m}{a^n}=a^{m-n}$	$a\neq0$ , $m,n\in\mathbb{Z}$
Raise a power to a power	$(a^m)^n=a^{mn}$	$a\neq0$ , $m,n\in\mathbb{Z}$
Raise a product to a power	$(ab)^m=a^m b^m$	$a,b\neq0$ , $m\in\mathbb{Z}$
Raise a quotient to a power	$\left(\frac ab\right)^m=\frac{a^m}{b^m}$	$a,b\neq0$ , $m\in\mathbb{Z}$
Multiply rational powers with equal denominators	$a^{m/n}a^{p/n}=a^{(m+p)/n}$	$a>0$ , $m,n,p\in\mathbb{Z}$ , $n\neq0$
Multiply 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 base stays the same when equal bases are multiplied or divided. Only the exponents are combined. By contrast, a power outside parentheses acts on the entire expression inside the parentheses.

Conditions for Exponent Rules

A zero base works for some positive exponents, but not for negative exponents because $0^{-1}$ would require division by zero. The integer rules therefore use $a\neq0$ as one consistent condition.
A quotient also requires a nonzero denominator.
For rational exponents, using $a>0$ keeps every root and power real and unambiguous. Some individual negative-base examples are real, but one unrestricted rule would not cover all rational exponents.

Each condition states when a property is valid. Check it before applying the property.

The Operation Determines the Exponent Rule

The table links each operation in the expression to the first step in simplifying it. Find the operation, then follow the matching row.

Pattern in the expression	First move
Same base multiplied	Add the exponents
Same base divided	Subtract the denominator exponent
Power outside a power	Multiply the exponents
Product or quotient inside parentheses	Distribute the outer exponent
Rational exponents with different denominators	Rewrite them with a common denominator

Simplifying Products Quotients and Powers

The examples below combine several rules in one expression. Apply one rule per step and keep the base unchanged. Write the intermediate line each time, because that line shows which rule you used.

Combining Products and Quotients

Each problem below combines multiplication and division of powers with the same base. Add the exponents of factors in the numerator, subtract the exponents of factors in the denominator, and keep the base unchanged.

Simplify $\frac{2^5\cdot2^3}{2^2}$ .
$\frac{2^5\cdot2^3}{2^2}=2^{5+3-2}$
Simplify $2^2\cdot2^3$ .

$2^2\cdot2^3=2^{2+3}=2^5=32$
Simplify $2^5\cdot2^2$ .

$2^5\cdot2^2=2^{5+2}=2^7=128$
Simplify $2^3\cdot2^7$ .

$2^3\cdot2^7=2^{3+7}=2^{10}=1024$
Simplify $\frac{2^8}{2^6}$ .

$\frac{2^8}{2^6}=2^{8-6}=2^2=4$
Simplify $\frac{2^{10}}{2^3}$ .

$\frac{2^{10}}{2^3}=2^{10-3}=2^7=128$
Simplify $\frac{2^6}{2^4}$ .

$\frac{2^6}{2^4}=2^{6-4}=2^2=4$

Powers of Powers

Each expression below raises a power to another power. Multiply the two exponents and keep the base unchanged, then check your result against the numeric value in the answer.

Simplify $(2^3)^3$ .

$(2^3)^3=2^{3\cdot3}=2^9=512$
Simplify $(2^4)^2$ .

$(2^4)^2=2^{4\cdot2}=2^8=256$
Simplify $(2^2)^5$ .

$(2^2)^5=2^{2\cdot5}=2^{10}=1024$

Rational Exponents

Simplify $2^{1/3}2^{2/3}$ .
$2^{1/3}2^{2/3}=2^{(1+2)/3}$
Simplify $3^{1/2}3^{1/3}$ .
$3^{1/2}3^{1/3}=3^{3/6+2/6}$

In each example, the operation and the arrangement of the bases determine which property applies. Reading that structure first is more reliable than trying to recall a formula from memory.