Precalculus Examples

$\frac{\sqrt{x} - \frac{1}{2 \sqrt{x}}}{\sqrt{x}}$

Step 1

Simplify the numerator.

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Step 1.1

Multiply $\frac{1}{2 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{x} - (\frac{1}{2 \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})}{\sqrt{x}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{2 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{x} - \frac{\sqrt{x}}{2 \sqrt{x} \sqrt{x}}}{\sqrt{x}}$

Step 1.2.2

Move $\sqrt{x}$ .

$\frac{\sqrt{x} - \frac{\sqrt{x}}{2 (\sqrt{x} \sqrt{x})}}{\sqrt{x}}$

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

$\frac{\sqrt{x} - \frac{\sqrt{x}}{2 {\sqrt{x}}^{1 + 1}}}{\sqrt{x}}$

Step 1.2.6

Add $1$ and $1$ .

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

Step 1.2.7

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

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Step 1.2.7.1

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

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

Step 1.2.7.2

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

$\frac{\sqrt{x} - \frac{\sqrt{x}}{2 x^{\frac{1}{2} \cdot 2}}}{\sqrt{x}}$

Step 1.2.7.3

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

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

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

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

Step 1.2.7.4.2

Rewrite the expression.

$\frac{\sqrt{x} - \frac{\sqrt{x}}{2 x^{1}}}{\sqrt{x}}$

Step 1.2.7.5

Simplify.

$\frac{\sqrt{x} - \frac{\sqrt{x}}{2 x}}{\sqrt{x}}$

Step 1.3

To write $\sqrt{x}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{\sqrt{x} \cdot \frac{2 x}{2 x} - \frac{\sqrt{x}}{2 x}}{\sqrt{x}}$

Step 1.4

Combine $\sqrt{x}$ and $\frac{2 x}{2 x}$ .

$\frac{\frac{\sqrt{x} (2 x)}{2 x} - \frac{\sqrt{x}}{2 x}}{\sqrt{x}}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{x} (2 x) - \sqrt{x}}{2 x}}{\sqrt{x}}$

Step 1.6

Factor $\sqrt{x}$ out of $\sqrt{x} (2 x) - \sqrt{x}$ .

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Step 1.6.1

Factor $\sqrt{x}$ out of $- \sqrt{x}$ .

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

Step 1.6.2

Factor $\sqrt{x}$ out of $\sqrt{x} (2 x) + \sqrt{x} \cdot - 1$ .

$\frac{\frac{\sqrt{x} (2 x - 1)}{2 x}}{\sqrt{x}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{x} (2 x - 1)}{2 x} \cdot \frac{1}{\sqrt{x}}$

Step 3

Cancel the common factor of $\sqrt{x}$ .

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Step 3.1

Cancel the common factor.

$\frac{\sqrt{x} (2 x - 1)}{2 x} \cdot \frac{1}{\sqrt{x}}$

Step 3.2

Rewrite the expression.

$\frac{2 x - 1}{2 x}$