Simplify ((4^-2)/(4^-7))^(-1/5)

${(\frac{4^{- 2}}{4^{- 7}})}^{- \frac{1}{5}}$

Step 1

Move $4^{- 7}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(4^{- 2} \cdot 4^{7})}^{- \frac{1}{5}}$

Step 2

Multiply $4^{- 2}$ by $4^{7}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(4^{- 2 + 7})}^{- \frac{1}{5}}$

Add $- 2$ and $7$ .

${(4^{5})}^{- \frac{1}{5}}$

Step 3

Multiply the exponents in ${(4^{5})}^{- \frac{1}{5}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4^{5 (- \frac{1}{5})}$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$4^{5 (\frac{- 1}{5})}$

Cancel the common factor.

$4^{5 (\frac{- 1}{5})}$

Rewrite the expression.

$4^{- 1}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$