Algebra Examples

$\frac{x^{2} - 81}{\sqrt{x} - 3}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 9^{2}}{\sqrt{x} - 3}$

Step 1.2

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 $b = 9$ .

$\frac{(x + 9) (x - 9)}{\sqrt{x} - 3}$

Step 2

Multiply $\frac{(x + 9) (x - 9)}{\sqrt{x} - 3}$ by $\frac{\sqrt{x} + 3}{\sqrt{x} + 3}$ .

$\frac{(x + 9) (x - 9)}{\sqrt{x} - 3} \cdot \frac{\sqrt{x} + 3}{\sqrt{x} + 3}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{(x + 9) (x - 9)}{\sqrt{x} - 3}$ by $\frac{\sqrt{x} + 3}{\sqrt{x} + 3}$ .

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{(\sqrt{x} - 3) (\sqrt{x} + 3)}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{{\sqrt{x}}^{2} + \sqrt{x} \cdot 3 - 3 \sqrt{x} - 9}$

Step 3.3

Simplify.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{{(x^{\frac{1}{2}})}^{2} - 9}$

Step 3.3.2

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

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{x^{\frac{1}{2} \cdot 2} - 9}$

Step 3.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{x^{\frac{2}{2}} - 9}$

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{x^{\frac{2}{2}} - 9}$

Step 3.3.4.2

Rewrite the expression.

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{x^{1} - 9}$

Step 3.3.5

Simplify.

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{x - 9}$

Step 3.4

Cancel the common factor of $x - 9$ .

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Step 3.4.1

Cancel the common factor.

$\frac{(x + 9) (x - 9) (\sqrt{x} + 3)}{x - 9}$

Step 3.4.2

Divide $(x + 9) (\sqrt{x} + 3)$ by $1$ .

$(x + 9) (\sqrt{x} + 3)$

Step 4

Expand $(x + 9) (\sqrt{x} + 3)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$x (\sqrt{x} + 3) + 9 (\sqrt{x} + 3)$

Step 4.2

Apply the distributive property.

$x \sqrt{x} + x \cdot 3 + 9 (\sqrt{x} + 3)$

Step 4.3

Apply the distributive property.

$x \sqrt{x} + x \cdot 3 + 9 \sqrt{x} + 9 \cdot 3$

Step 5

Simplify each term.

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Step 5.1

Move $3$ to the left of $x$ .

$x \sqrt{x} + 3 \cdot x + 9 \sqrt{x} + 9 \cdot 3$

Step 5.2

Multiply $9$ by $3$ .

$x \sqrt{x} + 3 x + 9 \sqrt{x} + 27$