MULTI-STEP EXPONENT LAW PRACTICE
LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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In this exercise you will practice with the exponent laws, all mixed-up.

These problems require the application of more than one exponent law.
For simpler problems, see One-Step Exponent Law Practice.

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
EXAMPLES:
$\displaystyle \cssId{s42}{\left(\frac{1}{x^2}\right)^3} \cssId{s43}{= (x^{-2})^3} \cssId{s44}{= x^{-2\,\cdot\, 3}} \cssId{s45}{= x^{-6}} \cssId{s46}{= x^p}$ where $\,p = -6$
$\displaystyle \cssId{s48}{\left(\frac{x^2}{x^3}\right)^5} \cssId{s49}{= (x^{2-3})^5} \cssId{s50}{= (x^{-1})^5} \cssId{s51}{= x^{-1\,\cdot\, 5}} \cssId{s52}{= x^{-5}} \cssId{s53}{= x^p} $ where $p = -5$
$\cssId{s55}{(x^2x^4)^{-1}} \cssId{s56}{= (x^{2+4})^{-1}} \cssId{s57}{= (x^6)^{-1}} \cssId{s58}{= x^{6\,\cdot\, -1}} \cssId{s59}{= x^{-6}} \cssId{s60}{= x^p}$ where $\,p = -6$
$\displaystyle \cssId{s62}{\frac{x^2x^{-3}}{x^5}} \cssId{s63}{= \frac{x^{2 + (-3)}}{x^5}} \cssId{s64}{= \frac{x^{-1}}{x^5}} \cssId{s65}{= x^{-1-5}} \cssId{s66}{= x^{-6}} \cssId{s67}{= x^p}$ where $\,p = -6$
$\displaystyle \cssId{s69}{\frac{x^2}{x^3x^4}} \cssId{s70}{= \frac{x^2}{x^{3+4}}} \cssId{s71}{= \frac{x^2}{x^7}} \cssId{s72}{= x^{2-7}} \cssId{s73}{= x^{-5}} \cssId{s74}{= x^p}$ where $\,p = -5$
$\displaystyle \cssId{s76}{\frac{(x^2)^3}{(x^{-1})^4}} \cssId{s77}{= \frac{x^{2\,\cdot\,3}}{x^{-1\,\cdot\,4}}} \cssId{s78}{= \frac{x^6}{x^{-4}}} \cssId{s79}{= x^{6-(-4)}} \cssId{s80}{= x^{10}} \cssId{s81}{= x^p}$ where $\,p = 10$
Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Practice with Radicals

 
 
Simplify:
    
(an even number, please)