Simplify ((9a^5t^-1)/(at^5))^2

${(\frac{9 a^{5} t^{- 1}}{a t^{5}})}^{2}$

Step 1

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

${(\frac{9 a^{5}}{a t^{5} t})}^{2}$

Step 2

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

Move $t$ .

${(\frac{9 a^{5}}{a (t \cdot t^{5})})}^{2}$

Multiply $t$ by $t^{5}$ .

Raise $t$ to the power of $1$ .

${(\frac{9 a^{5}}{a (t^{1} t^{5})})}^{2}$

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

${(\frac{9 a^{5}}{a t^{1 + 5}})}^{2}$

Add $1$ and $5$ .

${(\frac{9 a^{5}}{a t^{6}})}^{2}$

Step 3

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

Factor $a$ out of $9 a^{5}$ .

${(\frac{a (9 a^{4})}{a t^{6}})}^{2}$

Cancel the common factors.

Factor $a$ out of $a t^{6}$ .

${(\frac{a (9 a^{4})}{a (t^{6})})}^{2}$

Cancel the common factor.

${(\frac{a (9 a^{4})}{a t^{6}})}^{2}$

Rewrite the expression.

${(\frac{9 a^{4}}{t^{6}})}^{2}$

Step 4

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

Apply the product rule to $\frac{9 a^{4}}{t^{6}}$ .

$\frac{{(9 a^{4})}^{2}}{{(t^{6})}^{2}}$

Apply the product rule to $9 a^{4}$ .

$\frac{9^{2} {(a^{4})}^{2}}{{(t^{6})}^{2}}$

Step 5

Simplify the numerator.

Raise $9$ to the power of $2$ .

$\frac{81 {(a^{4})}^{2}}{{(t^{6})}^{2}}$

Multiply the exponents in ${(a^{4})}^{2}$ .

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

$\frac{81 a^{4 \cdot 2}}{{(t^{6})}^{2}}$

Multiply $4$ by $2$ .

$\frac{81 a^{8}}{{(t^{6})}^{2}}$

Step 6

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

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

$\frac{81 a^{8}}{t^{6 \cdot 2}}$

Multiply $6$ by $2$ .

$\frac{81 a^{8}}{t^{12}}$

Simplify ((9a^5t^-1)/(at^5))^2

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