Pre-Algebra Examples

${(\frac{4 a^{3} t^{2}}{9 a t^{6}})}^{- 3}$

Step 1

Cancel the common factor of $a^{3}$ and $a$ .

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Step 1.1

Factor $a$ out of $4 a^{3} t^{2}$ .

${(\frac{a (4 a^{2} t^{2})}{9 a t^{6}})}^{- 3}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $a$ out of $9 a t^{6}$ .

${(\frac{a (4 a^{2} t^{2})}{a (9 t^{6})})}^{- 3}$

Step 1.2.2

Cancel the common factor.

${(\frac{a (4 a^{2} t^{2})}{a (9 t^{6})})}^{- 3}$

Step 1.2.3

Rewrite the expression.

${(\frac{4 a^{2} t^{2}}{9 t^{6}})}^{- 3}$

Step 2

Cancel the common factor of $t^{2}$ and $t^{6}$ .

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Step 2.1

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

${(\frac{t^{2} (4 a^{2})}{9 t^{6}})}^{- 3}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $t^{2}$ out of $9 t^{6}$ .

${(\frac{t^{2} (4 a^{2})}{t^{2} (9 t^{4})})}^{- 3}$

Step 2.2.2

Cancel the common factor.

${(\frac{t^{2} (4 a^{2})}{t^{2} (9 t^{4})})}^{- 3}$

Step 2.2.3

Rewrite the expression.

${(\frac{4 a^{2}}{9 t^{4}})}^{- 3}$

Step 3

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

${(\frac{9 t^{4}}{4 a^{2}})}^{3}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{9 t^{4}}{4 a^{2}}$ .

$\frac{{(9 t^{4})}^{3}}{{(4 a^{2})}^{3}}$

Step 4.2

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

$\frac{9^{3} {(t^{4})}^{3}}{{(4 a^{2})}^{3}}$

Step 4.3

Apply the product rule to $4 a^{2}$ .

$\frac{9^{3} {(t^{4})}^{3}}{4^{3} {(a^{2})}^{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $9$ to the power of $3$ .

$\frac{729 {(t^{4})}^{3}}{4^{3} {(a^{2})}^{3}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{729 t^{4 \cdot 3}}{4^{3} {(a^{2})}^{3}}$

Step 5.2.2

Multiply $4$ by $3$ .

$\frac{729 t^{12}}{4^{3} {(a^{2})}^{3}}$

Step 6

Simplify the denominator.

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Step 6.1

Raise $4$ to the power of $3$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $3$ .

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