Simplify ((-2t^5)/(at^2))^4

${(\frac{- 2 t^{5}}{a t^{2}})}^{4}$

Step 1

Cancel the common factor of $t^{5}$ and $t^{2}$ .

Factor $t^{2}$ out of $- 2 t^{5}$ .

${(\frac{t^{2} (- 2 t^{3})}{a t^{2}})}^{4}$

Cancel the common factors.

Factor $t^{2}$ out of $a t^{2}$ .

${(\frac{t^{2} (- 2 t^{3})}{t^{2} a})}^{4}$

Cancel the common factor.

${(\frac{t^{2} (- 2 t^{3})}{t^{2} a})}^{4}$

Rewrite the expression.

${(\frac{- 2 t^{3}}{a})}^{4}$

Step 2

Move the negative in front of the fraction.

${(- \frac{(2) t^{3}}{a})}^{4}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{2 t^{3}}{a}$ .

${(- 1)}^{4} {(\frac{2 t^{3}}{a})}^{4}$

Apply the product rule to $\frac{2 t^{3}}{a}$ .

${(- 1)}^{4} \frac{{(2 t^{3})}^{4}}{a^{4}}$

Apply the product rule to $2 t^{3}$ .

${(- 1)}^{4} \frac{2^{4} {(t^{3})}^{4}}{a^{4}}$

Step 4

Simplify the expression.

Raise $- 1$ to the power of $4$ .

$1 \frac{2^{4} {(t^{3})}^{4}}{a^{4}}$

Multiply $\frac{2^{4} {(t^{3})}^{4}}{a^{4}}$ by $1$ .

$\frac{2^{4} {(t^{3})}^{4}}{a^{4}}$

Step 5

Simplify the numerator.

Raise $2$ to the power of $4$ .

$\frac{16 {(t^{3})}^{4}}{a^{4}}$

Multiply the exponents in ${(t^{3})}^{4}$ .

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

$\frac{16 t^{3 \cdot 4}}{a^{4}}$

Multiply $3$ by $4$ .

$\frac{16 t^{12}}{a^{4}}$