Trigonometry Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

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

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Simplify.

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

Step 1.3

Rewrite $\sqrt{\frac{\sqrt{x}}{x}}$ as $\frac{\sqrt{\sqrt{x}}}{\sqrt{x}}$ .

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

Step 1.4

Rewrite $\sqrt{\sqrt{x}}$ as $\sqrt[4]{x}$ .

$\frac{1}{\frac{\sqrt[4]{x}}{\sqrt{x}}}$

Step 1.5

Multiply $\frac{\sqrt[4]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{1}{\frac{\sqrt[4]{x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{\sqrt[4]{x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{\sqrt{x} \sqrt{x}}}$

Step 1.6.2

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

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}}$

Step 1.6.3

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

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}}$

Step 1.6.4

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

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}}$

Step 1.6.5

Add $1$ and $1$ .

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{{\sqrt{x}}^{2}}}$

Step 1.6.6

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

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Step 1.6.6.1

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

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}}$

Step 1.6.6.2

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

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}}$

Step 1.6.6.3

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

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{x^{\frac{2}{2}}}}$

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{x^{\frac{2}{2}}}}$

Step 1.6.6.4.2

Rewrite the expression.

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{x^{1}}}$

Step 1.6.6.5

Simplify.

$\frac{1}{\frac{\sqrt[4]{x} \sqrt{x}}{x}}$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Rewrite the expression using the least common index of $4$ .

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Step 1.7.1.1

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

$\frac{1}{\frac{\sqrt[4]{x} x^{\frac{1}{2}}}{x}}$

Step 1.7.1.2

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{2}{4}}$ .

$\frac{1}{\frac{\sqrt[4]{x} x^{\frac{2}{4}}}{x}}$

Step 1.7.1.3

Rewrite $x^{\frac{2}{4}}$ as $\sqrt[4]{x^{2}}$ .

$\frac{1}{\frac{\sqrt[4]{x} \sqrt[4]{x^{2}}}{x}}$

Step 1.7.2

Combine using the product rule for radicals.

$\frac{1}{\frac{\sqrt[4]{x \cdot x^{2}}}{x}}$

Step 1.7.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.7.3.1

Multiply $x$ by $x^{2}$ .

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Step 1.7.3.1.1

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

$\frac{1}{\frac{\sqrt[4]{x^{1} x^{2}}}{x}}$

Step 1.7.3.1.2

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

$\frac{1}{\frac{\sqrt[4]{x^{1 + 2}}}{x}}$

Step 1.7.3.2

Add $1$ and $2$ .

$\frac{1}{\frac{\sqrt[4]{x^{3}}}{x}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{x}{\sqrt[4]{x^{3}}}$

Step 3

Multiply $\frac{x}{\sqrt[4]{x^{3}}}$ by $1$ .

$\frac{x}{\sqrt[4]{x^{3}}}$

Step 4

Multiply $\frac{x}{\sqrt[4]{x^{3}}}$ by $\frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$ .

$\frac{x}{\sqrt[4]{x^{3}}} \cdot \frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$

Step 5

Simplify terms.

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Step 5.1

Combine and simplify the denominator.

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Step 5.1.1

Multiply $\frac{x}{\sqrt[4]{x^{3}}}$ by $\frac{{\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{3}}$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{\sqrt[4]{x^{3}} {\sqrt[4]{x^{3}}}^{3}}$

Step 5.1.2

Raise $\sqrt[4]{x^{3}}$ to the power of $1$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{1} {\sqrt[4]{x^{3}}}^{3}}$

Step 5.1.3

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

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{1 + 3}}$

Step 5.1.4

Add $1$ and $3$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{{\sqrt[4]{x^{3}}}^{4}}$

Step 5.1.5

Rewrite ${\sqrt[4]{x^{3}}}^{4}$ as $x^{3}$ .

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Step 5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{{(x^{\frac{3}{4}})}^{4}}$

Step 5.1.5.2

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

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{\frac{3}{4} \cdot 4}}$

Step 5.1.5.3

Combine $\frac{3}{4}$ and $4$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{\frac{3 \cdot 4}{4}}}$

Step 5.1.5.4

Multiply $3$ by $4$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{\frac{12}{4}}}$

Step 5.1.5.5

Cancel the common factor of $12$ and $4$ .

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Step 5.1.5.5.1

Factor $4$ out of $12$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{\frac{4 \cdot 3}{4}}}$

Step 5.1.5.5.2

Cancel the common factors.

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Step 5.1.5.5.2.1

Factor $4$ out of $4$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{\frac{4 \cdot 3}{4 (1)}}}$

Step 5.1.5.5.2.2

Cancel the common factor.

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{\frac{4 \cdot 3}{4 \cdot 1}}}$

Step 5.1.5.5.2.3

Rewrite the expression.

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{\frac{3}{1}}}$

Step 5.1.5.5.2.4

Divide $3$ by $1$ .

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x^{3}}$

Step 5.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 5.2.1

Factor $x$ out of $x {\sqrt[4]{x^{3}}}^{3}$ .

$\frac{x ({\sqrt[4]{x^{3}}}^{3})}{x^{3}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $x$ out of $x^{3}$ .

$\frac{x ({\sqrt[4]{x^{3}}}^{3})}{x \cdot x^{2}}$

Step 5.2.2.2

Cancel the common factor.

$\frac{x {\sqrt[4]{x^{3}}}^{3}}{x \cdot x^{2}}$

Step 5.2.2.3

Rewrite the expression.

$\frac{{\sqrt[4]{x^{3}}}^{3}}{x^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[4]{x^{3}}}^{3}$ as $\sqrt[4]{{(x^{3})}^{3}}$ .

$\frac{\sqrt[4]{{(x^{3})}^{3}}}{x^{2}}$

Step 6.2

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 6.2.1

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

$\frac{\sqrt[4]{x^{3 \cdot 3}}}{x^{2}}$

Step 6.2.2

Multiply $3$ by $3$ .

$\frac{\sqrt[4]{x^{9}}}{x^{2}}$

Step 6.3

Rewrite $x^{9}$ as ${(x^{2})}^{4} x$ .

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Step 6.3.1

Factor out $x^{8}$ .

$\frac{\sqrt[4]{x^{8} x}}{x^{2}}$

Step 6.3.2

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

$\frac{\sqrt[4]{{(x^{2})}^{4} x}}{x^{2}}$

Step 6.4

Pull terms out from under the radical.

$\frac{x^{2} \sqrt[4]{x}}{x^{2}}$

Step 7

Cancel the common factor of $x^{2}$ .

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Step 7.1

Cancel the common factor.

$\frac{x^{2} \sqrt[4]{x}}{x^{2}}$

Step 7.2

Divide $\sqrt[4]{x}$ by $1$ .

$\sqrt[4]{x}$