Simplify ((-2t^4)/(at))^6

${(\frac{- 2 t^{4}}{a t})}^{6}$

Cancel the common factor of $t^{4}$ and $t$ .

Factor $t$ out of $- 2 t^{4}$ .

${(\frac{t (- 2 t^{3})}{a t})}^{6}$

Cancel the common factors.

Factor $t$ out of $a t$ .

${(\frac{t (- 2 t^{3})}{t a})}^{6}$

Cancel the common factor.

${(\frac{t (- 2 t^{3})}{t a})}^{6}$

Rewrite the expression.

${(\frac{- 2 t^{3}}{a})}^{6}$

Move the negative in front of the fraction.

${(- \frac{(2) t^{3}}{a})}^{6}$

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

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

${(- 1)}^{6} {(\frac{2 t^{3}}{a})}^{6}$

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

${(- 1)}^{6} \frac{{(2 t^{3})}^{6}}{a^{6}}$

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

${(- 1)}^{6} \frac{2^{6} {(t^{3})}^{6}}{a^{6}}$

Simplify the expression.

Raise $- 1$ to the power of $6$ .

$1 \frac{2^{6} {(t^{3})}^{6}}{a^{6}}$

Multiply $\frac{2^{6} {(t^{3})}^{6}}{a^{6}}$ by $1$ .

$\frac{2^{6} {(t^{3})}^{6}}{a^{6}}$

Simplify the numerator.

Raise $2$ to the power of $6$ .

$\frac{64 {(t^{3})}^{6}}{a^{6}}$

Multiply the exponents in ${(t^{3})}^{6}$ .

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

$\frac{64 t^{3 \cdot 6}}{a^{6}}$

Multiply $3$ by $6$ .

$\frac{64 t^{18}}{a^{6}}$