Calculus Examples

$\frac{\sqrt{x^{2} + 22}}{5 \sqrt{21} - \frac{4 \sqrt{x}}{x}}$

Step 1

Simplify the denominator.

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Step 1.1

To write $5 \sqrt{21}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{5 \sqrt{21} x}{x} - \frac{4 \sqrt{x}}{x}}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{\sqrt{x^{2} + 22}}{\frac{5 \sqrt{21} x - 4 \sqrt{x}}{x}}$

Step 1.3

Rewrite $\frac{5 \sqrt{21} x - 4 \sqrt{x}}{x}$ in a factored form.

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{5 \cdot 21^{\frac{1}{2}} x - 4 \sqrt{x}}{x}}$

Step 1.3.2

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

$\frac{\sqrt{x^{2} + 22}}{\frac{5 \cdot 21^{\frac{1}{2}} x - 4 x^{\frac{1}{2}}}{x}}$

Step 1.3.3

Factor $x^{\frac{1}{2}}$ out of $5 \cdot 21^{\frac{1}{2}} x - 4 x^{\frac{1}{2}}$ .

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Step 1.3.3.1

Factor $x^{\frac{1}{2}}$ out of $5 \cdot 21^{\frac{1}{2}} x$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{x^{\frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}}) - 4 x^{\frac{1}{2}}}{x}}$

Step 1.3.3.2

Factor $x^{\frac{1}{2}}$ out of $- 4 x^{\frac{1}{2}}$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{x^{\frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot - 4}{x}}$

Step 1.3.3.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot - 4$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{x^{\frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)}{x}}$

Step 1.3.4

Reduce the expression by cancelling the common factors.

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Step 1.3.4.1

Reduce the expression $\frac{x^{\frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)}{x}$ by cancelling the common factors.

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Step 1.3.4.1.1

Factor $x$ out of $x^{\frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{x (x^{- \frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4))}{x}}$

Step 1.3.4.1.2

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

$\frac{\sqrt{x^{2} + 22}}{\frac{x (x^{- \frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4))}{x^{1}}}$

Step 1.3.4.1.3

Factor $x$ out of $x^{1}$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{x (x^{- \frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4))}{x \cdot 1}}$

Step 1.3.4.1.4

Cancel the common factor.

$\frac{\sqrt{x^{2} + 22}}{\frac{x (x^{- \frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4))}{x \cdot 1}}$

Step 1.3.4.1.5

Rewrite the expression.

$\frac{\sqrt{x^{2} + 22}}{\frac{x^{- \frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)}{1}}$

Step 1.3.4.2

Divide $x^{- \frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)$ by $1$ .

$\frac{\sqrt{x^{2} + 22}}{x^{- \frac{1}{2}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)}$

Step 1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt{x^{2} + 22}}{\frac{1}{x^{\frac{1}{2}}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)}$

Step 2

Factor $\frac{1}{x^{\frac{1}{2}}}$ out of $\frac{\sqrt{x^{2} + 22}}{\frac{1}{x^{\frac{1}{2}}} (5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4)}$ .

$x^{\frac{1}{2}} \frac{\sqrt{x^{2} + 22}}{5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4}$

Step 3

Combine $x^{\frac{1}{2}}$ and $\frac{\sqrt{x^{2} + 22}}{5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4}$ .

$\frac{x^{\frac{1}{2}} \sqrt{x^{2} + 22}}{5 \cdot 21^{\frac{1}{2}} x^{\frac{1}{2}} - 4}$