Simplify -(5t^5)/((5t^-5)^2)

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

Simplify the denominator.

Apply the product rule to $5 t^{- 5}$ .

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

Raise $5$ to the power of $2$ .

$- \frac{5 t^{5}}{25 {(t^{- 5})}^{2}}$

Multiply the exponents in ${(t^{- 5})}^{2}$ .

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

$- \frac{5 t^{5}}{25 t^{- 5 \cdot 2}}$

Multiply $- 5$ by $2$ .

$- \frac{5 t^{5}}{25 t^{- 10}}$

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

$- \frac{5 t^{5}}{25 \frac{1}{t^{10}}}$

Combine $25$ and $\frac{1}{t^{10}}$ .

$- \frac{5 t^{5}}{\frac{25}{t^{10}}}$

Multiply the numerator by the reciprocal of the denominator.

$- (5 t^{5} \frac{t^{10}}{25})$

Cancel the common factor of $5$ .

Factor $5$ out of $5 t^{5}$ .

$- (5 (t^{5}) \frac{t^{10}}{25})$

Factor $5$ out of $25$ .

$- (5 (t^{5}) \frac{t^{10}}{5 (5)})$

Cancel the common factor.

$- (5 t^{5} \frac{t^{10}}{5 \cdot 5})$

Rewrite the expression.

$- (t^{5} \frac{t^{10}}{5})$

Combine $t^{5}$ and $\frac{t^{10}}{5}$ .

$- \frac{t^{5} t^{10}}{5}$

Multiply $t^{5}$ by $t^{10}$ by adding the exponents.

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

$- \frac{t^{5 + 10}}{5}$

Add $5$ and $10$ .

$- \frac{t^{15}}{5}$