Question

write the product of the expression 5^4 * 5^-7 using a positive exponent, then write the product using a negative exponent

Answer 1

Answer with positive exponent = $\left(\frac{1}{5}\right)^3$ or  $\frac{1}{5^3}$

Answer with negative exponent = $5^{-3}$

===========================================================

Explanation:

The rule we use is

$a^b*a^c = a^{b+c}$

If we multiply two exponential expressions with the same base, then we add the exponents.

The base for each is 5. The exponents 4 and -7 add to -3.

This means

$5^4*5^{-7} = 5^{4+(-7)} = 5^{-3}$

To convert to a positive exponent, we apply the reciprocal to the base. We go from 5, aka 5/1, to 1/5.

So, $5^{-3} = \left(\frac{1}{5}\right)^3 = \frac{1^3}{5^3} = \frac{1}{5^3}$