# How do you write the product of the expression 5^4*5^-7 using a positive exponent?

Nov 20, 2016

$\frac{1}{5} ^ 3$

#### Explanation:

Using the $\textcolor{b l u e}{\text{laws of exponents}}$

color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(a^mxxa^n=a^(m+n)" and " a^-mhArr1/a^m)color(white)(2/2)|)))

$\Rightarrow {5}^{4} \times {5}^{-} 7 = {5}^{-} 3 = \frac{1}{5} ^ 3$

Nov 20, 2016

$\frac{1}{5} ^ 3$

#### Explanation:

Another approach is to get rid of the negative index first..

Recall: ${x}^{-} m = \frac{1}{x} ^ \left(+ m\right) \text{ and } {x}^{m} / {x}^{n} = \frac{1}{x} ^ \left(n - m\right) \mathmr{if} n > m$

${5}^{4} \times {5}^{-} 7$

=${5}^{4} \times \frac{1}{5} ^ 7$

=$\frac{1}{5} ^ \left(7 - 4\right)$

=$\frac{1}{5} ^ 3$