Basic Math Examples

${(3 a^{4})}^{- 4} {({(\frac{1}{9 a^{8}})}^{- 1})}^{2}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(3 a^{4})}^{4}} {({(\frac{1}{9 a^{8}})}^{- 1})}^{2}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $3 a^{4}$ .

$\frac{1}{3^{4} {(a^{4})}^{4}} {({(\frac{1}{9 a^{8}})}^{- 1})}^{2}$

Step 2.2

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

$\frac{1}{81 {(a^{4})}^{4}} {({(\frac{1}{9 a^{8}})}^{- 1})}^{2}$

Step 2.3

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

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Step 2.3.1

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

$\frac{1}{81 a^{4 \cdot 4}} {({(\frac{1}{9 a^{8}})}^{- 1})}^{2}$

Step 2.3.2

Multiply $4$ by $4$ .

$\frac{1}{81 a^{16}} {({(\frac{1}{9 a^{8}})}^{- 1})}^{2}$

Step 3

Multiply the exponents in ${({(\frac{1}{9 a^{8}})}^{- 1})}^{2}$ .

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Step 3.1

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

$\frac{1}{81 a^{16}} {(\frac{1}{9 a^{8}})}^{- 1 \cdot 2}$

Step 3.2

Multiply $- 1$ by $2$ .

$\frac{1}{81 a^{16}} {(\frac{1}{9 a^{8}})}^{- 2}$

Step 4

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

$\frac{1}{81 a^{16}} {(9 a^{8})}^{2}$

Step 5

Simplify the expression.

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Step 5.1

Apply the product rule to $9 a^{8}$ .

$\frac{1}{81 a^{16}} (9^{2} {(a^{8})}^{2})$

Step 5.2

Rewrite using the commutative property of multiplication.

$9^{2} \frac{1}{81 a^{16}} {(a^{8})}^{2}$

Step 5.3

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

$81 \frac{1}{81 a^{16}} {(a^{8})}^{2}$

Step 6

Cancel the common factor of $81$ .

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Step 6.1

Factor $81$ out of $81 a^{16}$ .

$81 \frac{1}{81 (a^{16})} {(a^{8})}^{2}$

Step 6.2

Cancel the common factor.

$81 \frac{1}{81 a^{16}} {(a^{8})}^{2}$

Step 6.3

Rewrite the expression.

$\frac{1}{a^{16}} {(a^{8})}^{2}$

Step 7

Multiply the exponents in ${(a^{8})}^{2}$ .

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Step 7.1

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

$\frac{1}{a^{16}} a^{8 \cdot 2}$

Step 7.2

Multiply $8$ by $2$ .

$\frac{1}{a^{16}} a^{16}$

Step 8

Cancel the common factor of $a^{16}$ .

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Step 8.1

Cancel the common factor.

$\frac{1}{a^{16}} a^{16}$

Step 8.2

Rewrite the expression.

$1$