Simplify ((8a^4t)/(27at^7))^-2

${(\frac{8 a^{4} t}{27 a t^{7}})}^{- 2}$

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

Factor $a$ out of $8 a^{4} t$ .

${(\frac{a (8 a^{3} t)}{27 a t^{7}})}^{- 2}$

Cancel the common factors.

Factor $a$ out of $27 a t^{7}$ .

${(\frac{a (8 a^{3} t)}{a (27 t^{7})})}^{- 2}$

Cancel the common factor.

${(\frac{a (8 a^{3} t)}{a (27 t^{7})})}^{- 2}$

Rewrite the expression.

${(\frac{8 a^{3} t}{27 t^{7}})}^{- 2}$

Cancel the common factor of $t$ and $t^{7}$ .

Factor $t$ out of $8 a^{3} t$ .

${(\frac{t (8 a^{3})}{27 t^{7}})}^{- 2}$

Cancel the common factors.

Factor $t$ out of $27 t^{7}$ .

${(\frac{t (8 a^{3})}{t (27 t^{6})})}^{- 2}$

Cancel the common factor.

${(\frac{t (8 a^{3})}{t (27 t^{6})})}^{- 2}$

Rewrite the expression.

${(\frac{8 a^{3}}{27 t^{6}})}^{- 2}$

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{27 t^{6}}{8 a^{3}})}^{2}$

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

Apply the product rule to $\frac{27 t^{6}}{8 a^{3}}$ .

$\frac{{(27 t^{6})}^{2}}{{(8 a^{3})}^{2}}$

Apply the product rule to $27 t^{6}$ .

$\frac{27^{2} {(t^{6})}^{2}}{{(8 a^{3})}^{2}}$

Apply the product rule to $8 a^{3}$ .

$\frac{27^{2} {(t^{6})}^{2}}{8^{2} {(a^{3})}^{2}}$

Simplify the numerator.

Raise $27$ to the power of $2$ .

$\frac{729 {(t^{6})}^{2}}{8^{2} {(a^{3})}^{2}}$

Multiply the exponents in ${(t^{6})}^{2}$ .

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

$\frac{729 t^{6 \cdot 2}}{8^{2} {(a^{3})}^{2}}$

Multiply $6$ by $2$ .

$\frac{729 t^{12}}{8^{2} {(a^{3})}^{2}}$

Simplify the denominator.

Raise $8$ to the power of $2$ .

$\frac{729 t^{12}}{64 {(a^{3})}^{2}}$

Multiply the exponents in ${(a^{3})}^{2}$ .

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

$\frac{729 t^{12}}{64 a^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{729 t^{12}}{64 a^{6}}$

Simplify ((8a^4t)/(27at^7))^-2

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