Precalculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 1.3

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

$\frac{1}{\frac{2 + \sqrt{x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}}$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

Add $1$ and $1$ .

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

Step 1.4.6

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

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Step 1.4.6.1

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

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

Step 1.4.6.2

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

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

Step 1.4.6.3

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

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

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

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

Step 1.4.6.4.2

Rewrite the expression.

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

Step 1.4.6.5

Simplify.

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 1.6.1

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

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

Step 1.6.2

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

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

Step 1.6.3

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

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

Step 1.6.4

Add $1$ and $1$ .

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

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

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

Step 1.7.4

Cancel the common factor of $2$ .

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Step 1.7.4.1

Cancel the common factor.

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

Step 1.7.4.2

Rewrite the expression.

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

Step 1.7.5

Simplify.

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

Step 1.8

Simplify the numerator.

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Step 1.8.1

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

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

Step 1.8.2

Factor $x^{\frac{1}{2}}$ out of $2 x^{\frac{1}{2}} + x$ .

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Step 1.8.2.1

Factor $x^{\frac{1}{2}}$ out of $2 x^{\frac{1}{2}}$ .

$\frac{1}{\frac{x^{\frac{1}{2}} \cdot 2 + x}{x}}$

Step 1.8.2.2

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

$\frac{1}{\frac{x^{\frac{1}{2}} \cdot 2 + x^{1}}{x}}$

Step 1.8.2.3

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

$\frac{1}{\frac{x^{\frac{1}{2}} \cdot 2 + x^{\frac{1}{2}} x^{\frac{1}{2}}}{x}}$

Step 1.8.2.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 2 + x^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{1}{\frac{x^{\frac{1}{2}} (2 + x^{\frac{1}{2}})}{x}}$

Step 1.9

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 1.10

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.10.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 1.10.1.1

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

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

Step 1.10.1.2

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

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

Step 1.10.2

Write $1$ as a fraction with a common denominator.

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

Step 1.10.3

Combine the numerators over the common denominator.

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

Step 1.10.4

Subtract $1$ from $2$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{x^{\frac{1}{2}}}{2 + x^{\frac{1}{2}}}$ by $1$ .

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