Pre-Algebra Examples

$f (x) = \frac{x - 7}{\sqrt{x - 7}}$

Step 1

Rewrite the function as an equation.

$y = \frac{x - 7}{\sqrt{x - 7}}$

Step 2

Simplify $\frac{x - 7}{\sqrt{x - 7}}$ .

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Step 2.1

Multiply $\frac{x - 7}{\sqrt{x - 7}}$ by $\frac{\sqrt{x - 7}}{\sqrt{x - 7}}$ .

$y = \frac{x - 7}{\sqrt{x - 7}} \cdot \frac{\sqrt{x - 7}}{\sqrt{x - 7}}$

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

Multiply $\frac{x - 7}{\sqrt{x - 7}}$ by $\frac{\sqrt{x - 7}}{\sqrt{x - 7}}$ .

$y = \frac{(x - 7) \sqrt{x - 7}}{\sqrt{x - 7} \sqrt{x - 7}}$

Step 2.2.2

Raise $\sqrt{x - 7}$ to the power of $1$ .

$y = \frac{(x - 7) \sqrt{x - 7}}{{\sqrt{x - 7}}^{1} \sqrt{x - 7}}$

Step 2.2.3

Raise $\sqrt{x - 7}$ to the power of $1$ .

$y = \frac{(x - 7) \sqrt{x - 7}}{{\sqrt{x - 7}}^{1} {\sqrt{x - 7}}^{1}}$

Step 2.2.4

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

$y = \frac{(x - 7) \sqrt{x - 7}}{{\sqrt{x - 7}}^{1 + 1}}$

Step 2.2.5

Add $1$ and $1$ .

$y = \frac{(x - 7) \sqrt{x - 7}}{{\sqrt{x - 7}}^{2}}$

Step 2.2.6

Rewrite ${\sqrt{x - 7}}^{2}$ as $x - 7$ .

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Step 2.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 7}$ as ${(x - 7)}^{\frac{1}{2}}$ .

$y = \frac{(x - 7) \sqrt{x - 7}}{{({(x - 7)}^{\frac{1}{2}})}^{2}}$

Step 2.2.6.2

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

$y = \frac{(x - 7) \sqrt{x - 7}}{{(x - 7)}^{\frac{1}{2} \cdot 2}}$

Step 2.2.6.3

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

$y = \frac{(x - 7) \sqrt{x - 7}}{{(x - 7)}^{\frac{2}{2}}}$

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

$y = \frac{(x - 7) \sqrt{x - 7}}{{(x - 7)}^{\frac{2}{2}}}$

Step 2.2.6.4.2

Rewrite the expression.

$y = \frac{(x - 7) \sqrt{x - 7}}{{(x - 7)}^{1}}$

Step 2.2.6.5

Simplify.

$y = \frac{(x - 7) \sqrt{x - 7}}{x - 7}$

Step 2.3

Cancel the common factor of $x - 7$ .

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Step 2.3.1

Cancel the common factor.

$y = \frac{(x - 7) \sqrt{x - 7}}{x - 7}$

Step 2.3.2

Divide $\sqrt{x - 7}$ by $1$ .

$y = \sqrt{x - 7}$