Trigonometry Examples

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

Step 1

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

$\frac{\sqrt{n}}{\sqrt{n} - \sqrt{x}} \cdot \frac{\sqrt{n} + \sqrt{x}}{\sqrt{n} + \sqrt{x}}$

Step 2

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

$\frac{\sqrt{n} (\sqrt{n} + \sqrt{x})}{(\sqrt{n} - \sqrt{x}) (\sqrt{n} + \sqrt{x})}$

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

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

Step 4.1.4.2

Rewrite the expression.

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

Step 4.1.5

Simplify.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Rewrite the expression.

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

Step 4.2.5

Simplify.

$\frac{\sqrt{n} (\sqrt{n} + \sqrt{x})}{n - x}$

Step 5

Apply the distributive property.

$\frac{\sqrt{n} \sqrt{n} + \sqrt{n} \sqrt{x}}{n - x}$

Step 6

Multiply $\sqrt{n} \sqrt{n}$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 7

Simplify terms.

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

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

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

Step 7.2.4

Cancel the common factor of $2$ .

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Step 7.2.4.1

Cancel the common factor.

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

Step 7.2.4.2

Rewrite the expression.

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

Step 7.2.5

Simplify.

$\frac{n + \sqrt{n x}}{n - x}$

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

Apply the product rule to $n x$ .

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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