Algebra Examples

$f (x) = x^{2} - 81$ $g (x) = {(x - 9)}^{- 1} (x + 9)$

Step 1

Replace the function designators with the actual functions in $f (x) \cdot (g (x))$ .

$(x^{2} - 81) \cdot ({(x - 9)}^{- 1} (x + 9))$

Step 2

Simplify.

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Step 2.1

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

$(x^{2} - 81) \cdot (\frac{1}{x - 9} (x + 9))$

Step 2.2

Combine fractions.

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Step 2.2.1

Multiply $\frac{1}{x - 9}$ by $x + 9$ .

$(x^{2} - 81) \cdot \frac{x + 9}{x - 9}$

Step 2.2.2

Multiply $x^{2} - 81$ by $\frac{x + 9}{x - 9}$ .

$\frac{(x^{2} - 81) (x + 9)}{x - 9}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Rewrite $81$ as $9^{2}$ .

$\frac{(x^{2} - 9^{2}) (x + 9)}{x - 9}$

Step 2.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$\frac{(x + 9) (x - 9) (x + 9)}{x - 9}$

Step 2.3.3

Combine exponents.

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Step 2.3.3.1

Raise $x + 9$ to the power of $1$ .

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

Step 2.3.3.2

Raise $x + 9$ to the power of $1$ .

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

Step 2.3.3.3

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

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

Step 2.3.3.4

Add $1$ and $1$ .

$\frac{{(x + 9)}^{2} (x - 9)}{x - 9}$

Step 2.4

Reduce the expression by cancelling the common factors.

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Step 2.4.1

Cancel the common factor of $x - 9$ .

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Step 2.4.1.1

Cancel the common factor.

$\frac{{(x + 9)}^{2} (x - 9)}{x - 9}$

Step 2.4.1.2

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

${(x + 9)}^{2}$

Step 2.4.2

Rewrite ${(x + 9)}^{2}$ as $(x + 9) (x + 9)$ .

$(x + 9) (x + 9)$

Step 2.5

Expand $(x + 9) (x + 9)$ using the FOIL Method.

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Step 2.5.1

Apply the distributive property.

$x (x + 9) + 9 (x + 9)$

Step 2.5.2

Apply the distributive property.

$x \cdot x + x \cdot 9 + 9 (x + 9)$

Step 2.5.3

Apply the distributive property.

$x \cdot x + x \cdot 9 + 9 x + 9 \cdot 9$

Step 2.6

Simplify and combine like terms.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 9 + 9 x + 9 \cdot 9$

Step 2.6.1.2

Move $9$ to the left of $x$ .

$x^{2} + 9 \cdot x + 9 x + 9 \cdot 9$

Step 2.6.1.3

Multiply $9$ by $9$ .

$x^{2} + 9 x + 9 x + 81$

Step 2.6.2

Add $9 x$ and $9 x$ .

$x^{2} + 18 x + 81$