Pre-Algebra Examples

$(\frac{9^{2}}{9^{- 12}}) (\frac{9^{- 3}}{{(- 9)}^{9}})$

Step 1

Combine.

$\frac{9^{2} \cdot 9^{- 3}}{9^{- 12} {(- 9)}^{9}}$

Step 2

Multiply $9^{2}$ by $9^{- 3}$ by adding the exponents.

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Step 2.1

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

$\frac{9^{2 - 3}}{9^{- 12} {(- 9)}^{9}}$

Step 2.2

Subtract $3$ from $2$ .

$\frac{9^{- 1}}{9^{- 12} {(- 9)}^{9}}$

Step 3

Move $9^{- 12}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{9^{- 1} \cdot 9^{12}}{{(- 9)}^{9}}$

Step 4

Multiply $9^{- 1}$ by $9^{12}$ by adding the exponents.

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Step 4.1

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

$\frac{9^{- 1 + 12}}{{(- 9)}^{9}}$

Step 4.2

Add $- 1$ and $12$ .

$\frac{9^{11}}{{(- 9)}^{9}}$

Step 5

Cancel the common factor of $9^{11}$ and ${(- 9)}^{9}$ .

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Step 5.1

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{{(- 1 (- 9))}^{11}}{{(- 9)}^{9}}$

Step 5.2

Apply the product rule to $- 1 (- 9)$ .

$\frac{{(- 1)}^{11} {(- 9)}^{11}}{{(- 9)}^{9}}$

Step 5.3

Raise $- 1$ to the power of $11$ .

$\frac{- 1 {(- 9)}^{11}}{{(- 9)}^{9}}$

Step 5.4

Factor ${(- 9)}^{9}$ out of $- 1 {(- 9)}^{11}$ .

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

Step 5.5

Cancel the common factors.

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Step 5.5.1

Multiply by $1$ .

$\frac{{(- 9)}^{9} (- 1 {(- 9)}^{2})}{{(- 9)}^{9} \cdot 1}$

Step 5.5.2

Cancel the common factor.

$\frac{{(- 9)}^{9} (- 1 {(- 9)}^{2})}{{(- 9)}^{9} \cdot 1}$

Step 5.5.3

Rewrite the expression.

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

Step 5.5.4

Divide $- 1 {(- 9)}^{2}$ by $1$ .

$- 1 {(- 9)}^{2}$

Step 6

Simplify the expression.

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Step 6.1

Rewrite $- 1 {(- 9)}^{2}$ as $- {(- 9)}^{2}$ .

$- {(- 9)}^{2}$

Step 6.2

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

$- 1 \cdot 81$

Step 6.3

Multiply $- 1$ by $81$ .

$- 81$