Exponent and Radical Rules — AP® Precalculus Notes

Unit 2 is built on powers: amounts that double, halve, or grow by $3\%$ a year. Every one of those models is written with exponents, and every Unit 2 manipulation uses the same few rules. This page collects them, with each step shown, so you can use them without a calculator.

The exponent rules

For a positive base $b$ and any exponents $m$ and $n$ :

Rule	In symbols	Example
Multiply, same base: add exponents	$b^m \cdot b^n = b^{m + n}$	$2^3 \cdot 2^4 = 2^7$
Divide, same base: subtract exponents	$\frac{b^m}{b^n} = b^{m - n}$	$\frac{5^6}{5^2} = 5^4$
Power of a power: multiply exponents	$(b^m)^n = b^{mn}$	$(x^3)^2 = x^6$
Power of a product	$(ab)^n = a^n b^n$	$(2x^3)^2 = 4x^6$
Zero exponent	$b^0 = 1$	$7^0 = 1$
Negative exponent: reciprocal	$b^{-n} = \frac{1}{b^n}$	$2^{-3} = \frac{1}{8}$
Rational exponent: root, then power	$b^{m/n} = (\sqrt[n]{b})^m$	$8^{2/3} = 2^2 = 4$

The most expensive slip is changing the base. $2^3 \cdot 2^4$ is $2^7$ , never $4^7$ : the base stays the same, and only the exponents combine.

Negative exponents

Why does $b^{-n}$ mean $\frac{1}{b^n}$ ? Follow the pattern down: $2^3 = 8$ , $2^2 = 4$ , $2^1 = 2$ , $2^0 = 1$ . Each step down divides by $2$ . One more step gives $2^{-1} = \frac{1}{2}$ , then $2^{-2} = \frac{1}{4}$ . A negative exponent makes a fraction, not a negative number.

Rational exponents and radicals

The exponent $\frac{1}{n}$ is an $n$ th root, because $(b^{1/n})^n = b^1 = b$ . So $9^{1/2} = \sqrt{9} = 3$ and $27^{1/3} = \sqrt[3]{27} = 3$ . In $b^{m/n}$ , the denominator is the root and the numerator is the power.

Graphs of y = x^(1/2) and y = x^(1/3) for x from 0 to 9, meeting at (0, 0) and (1, 1); the square root passes (4, 2) and (9, 3), and the cube root passes (8, 2). — **Figure 1.** $x^{1/2}$ and $x^{1/3}$ : the square root and the cube root. LevlPrep original diagram.

Worked example 1. Evaluate $16^{-3/4}$ .

Negative exponent first: $16^{-3/4} = \frac{1}{16^{3/4}}$ .

Root before power keeps the numbers small: $16^{1/4} = 2$ , because $2^4 = 16$ .

Then the power: $16^{3/4} = 2^3 = 8$ . So $16^{-3/4} = \frac{1}{8}$ .

Worked example 2. Write $\frac{\sqrt{x^3}}{x^{-1}}$ as a single power of $x$ , for $x > 0$ .

The root as a power: $\sqrt{x^3}$ $= (x^3)^{1/2}$ $= x^{3/2}$ .

Divide, same base, subtract exponents: $\frac{x^{3/2}}{x^{-1}}$ $= x^{3/2 - (-1)}$ $= x^{5/2}$ .

Matching bases to compare exponents

If two powers of the same base are equal, their exponents are equal. That lets you solve some equations with no calculator: rewrite both sides with one base.

Worked example 3. Solve $4^x = 32$ .

Both are powers of $2$ : $4 = 2^2$ and $32 = 2^5$ .

So $(2^2)^x = 2^5$ , which is $2^{2x} = 2^5$ .

Equal powers of $2$ have equal exponents: $2x = 5$ , so $x = \frac{5}{2}$ . Check: $4^{5/2}$ $= (\sqrt{4})^5$ $= 2^5$ $= 32$ .

In Unit 2 the same moves rewrite a model: $9^t = (3^2)^t = 3^{2t}$ , and $2^{t + 3}$ $= 2^t \cdot 2^3$ $= 8 \cdot 2^t$ .