audio read-through One-Step Exponent Law Practice

In this exercise you will practice with the exponent laws, all mixed-up.

These problems require only a single application of a single exponent law. For more advanced problems, see Multi-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

$\displaystyle x^2x^{-5} = x^p\,$   where   $\,p = \text{?}$
Answer: $p = -3$
$\displaystyle \frac{x^5}{x^3} = x^p\,$   where   $\,p = \text{?}$
Answer: $p = 2$
$\displaystyle (x^3)^2 = x^p\,$   where   $\,p = \text{?}$
Answer: $p = 6$
$\displaystyle \frac{1}{x^7} = x^p$   where   $\,p = \text{?}$
Answer: $p = -7$
$\displaystyle \frac{1}{x^{-7}} = x^p$   where $\,p = \text{?}$
Answer: $p = 7$

Practice