﻿ Practice with "Something to a Power to a Power, Multiply the Exponents"
PRACTICE WITH $\,(x^m)^n = x^{mn}$

by Dr. Carol JVF Burns (website creator)
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• PRACTICE (online exercises and printable worksheets)
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All the exponent laws are stated below, for completeness.
This web exercise gives practice with: $$\cssId{s6}{(x^m)^n = x^{mn}}$$ Here's the motivation for this exponent law: $$\cssId{s8}{(x^2)^3} \cssId{s9}{= (x^2)(x^2)(x^2)} \cssId{s10}{= \overset{\text{three piles, two in each}}{\overbrace{(x\cdot x)(x\cdot x)(x\cdot x)}}} \cssId{s11}{= x^6} \cssId{s12}{= x^{2\cdot 3}}$$

EXPONENT LAWS
Let $\,x\,$, $\,y\,$, $\,m\,$, and $\,n\,$ be real numbers, with the following exceptions:
• a base and exponent cannot simultaneously be zero (since $\,0^0\,$ is undefined);
• division by zero is not allowed;
• for non-integer exponents (like $\,\frac12\,$ or $\,0.4\,$), assume that bases are positive.
Then,
 $x^mx^n = x^{m+n}$ Verbalize: same base, things multiplied, add the exponents $\displaystyle \frac{x^m}{x^n} = x^{m-n}$ Verbalize: same base, things divided, subtract the exponents $(x^m)^n = x^{mn}$ Verbalize: something to a power, to a power; multiply the exponents $(xy)^m = x^my^m$ Verbalize: product to a power; each factor gets raised to the power $\displaystyle \left(\frac{x}{y}\right)^m = \frac{x^m}{y^m}$ Verbalize: fraction to a power; both numerator and denominator get raised to the power
EXAMPLE:
$(x^3)^2 = x^p\,$ where $\,p = \text{?}$
Answer: $p = 6$
Master the ideas from this section
Practice with $\,x^m/x^n = x^{m-n}$