Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Simplify.

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

Step 5

Cancel the common factor of $x - 9$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

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

$\sqrt{x} + 3$