Simplify (1/(t^-7*t^-1))^(5/16)

${(\frac{1}{t^{- 7} \cdot t^{- 1}})}^{\frac{5}{16}}$

Multiply $t^{- 7}$ by $t^{- 1}$ by adding the exponents.

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

${(\frac{1}{t^{- 7 - 1}})}^{\frac{5}{16}}$

Subtract $1$ from $- 7$ .

${(\frac{1}{t^{- 8}})}^{\frac{5}{16}}$

Move $t^{- 8}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(t^{8})}^{\frac{5}{16}}$

Multiply the exponents in ${(t^{8})}^{\frac{5}{16}}$ .

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

$t^{8 (\frac{5}{16})}$

Cancel the common factor of $8$ .

Factor $8$ out of $16$ .

$t^{8 \frac{5}{8 (2)}}$

Cancel the common factor.

$t^{8 \frac{5}{8 \cdot 2}}$

Rewrite the expression.

$t^{\frac{5}{2}}$