PRACTICE WITH $\,x^mx^n = x^{m+n}$

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
 

All the exponent laws are stated below, for completeness.
This web exercise gives practice with: $$ \cssId{s6}{x^mx^n = x^{m+n}} $$ Here's the motivation for this exponent law: $$ \cssId{s8}{x^2} \cssId{s9}{x^3} \cssId{s10}{= \overset{\text{two factors}}{\overbrace{x\cdot x}}} \cssId{s11}{\cdot \overset{\text{three factors}}{\overbrace{x\cdot x\cdot x}}} \cssId{s12}{\ = \ \overset{\text{five factors}}{\overbrace{x\cdot x\cdot x\cdot x\cdot x}}} \cssId{s13}{\ = \ x^5} \cssId{s14}{\ = \ x^{2+3}} $$

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:
$x^2x^{-5} = x^p\,$ where $\,p = \text{?}$
Answer: $p = -3$
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 $\,(x^m)^n = x^{mn}$

 
 
Simplify:
    
(an even number, please)