Trigonometry Examples

$\sqrt[4]{\frac{7^{2}}{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 1

Raise $7$ to the power of $2$ .

$\sqrt[4]{\frac{49}{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 2

Rewrite $\sqrt[4]{\frac{49}{m}}$ as $\frac{\sqrt[4]{49}}{\sqrt[4]{m}}$ .

$\frac{\sqrt[4]{49}}{\sqrt[4]{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 3.2

Rewrite $\sqrt[4]{7^{2}}$ as $\sqrt{\sqrt{7^{2}}}$ .

$\frac{\sqrt{\sqrt{7^{2}}}}{\sqrt[4]{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 3.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{7}}{\sqrt[4]{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 4

Multiply $\frac{\sqrt{7}}{\sqrt[4]{m}}$ by $\frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}}$ .

$\frac{\sqrt{7}}{\sqrt[4]{m}} \cdot \frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{7}}{\sqrt[4]{m}}$ by $\frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}}$ .

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{\sqrt[4]{m} {\sqrt[4]{m}}^{3}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.2

Raise $\sqrt[4]{m}$ to the power of $1$ .

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{1} {\sqrt[4]{m}}^{3}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.3

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

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{1 + 3}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.4

Add $1$ and $3$ .

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{4}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.5

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

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{m}$ as $m^{\frac{1}{4}}$ .

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{(m^{\frac{1}{4}})}^{4}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.5.2

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

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{1}{4} \cdot 4}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{4}{4}}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.5.4

Cancel the common factor of $4$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{4}{4}}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{1}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 5.5.5

Simplify.

$\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 6

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

$\frac{\sqrt{7} \sqrt[4]{m^{3}}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{7^{\frac{1}{2}} \sqrt[4]{m^{3}}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 7.1.2

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{2}{4}}$ .

$\frac{7^{\frac{2}{4}} \sqrt[4]{m^{3}}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 7.1.3

Rewrite $7^{\frac{2}{4}}$ as $\sqrt[4]{7^{2}}$ .

$\frac{\sqrt[4]{7^{2}} \sqrt[4]{m^{3}}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 7.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{7^{2} m^{3}}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 7.3

Raise $7$ to the power of $2$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $7$ to the power of $2$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt[4]{49}}{\sqrt[4]{m}}$

Step 8.2

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 8.3

Rewrite $\sqrt[4]{7^{2}}$ as $\sqrt{\sqrt{7^{2}}}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{\sqrt{7^{2}}}}{\sqrt[4]{m}}$

Step 8.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7}}{\sqrt[4]{m}}$

Step 9

Multiply $\frac{\sqrt{7}}{\sqrt[4]{m}}$ by $\frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7}}{\sqrt[4]{m}} \cdot \frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{\sqrt{7}}{\sqrt[4]{m}}$ by $\frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{\sqrt[4]{m} {\sqrt[4]{m}}^{3}}$

Step 10.2

Raise $\sqrt[4]{m}$ to the power of $1$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{1} {\sqrt[4]{m}}^{3}}$

Step 10.3

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

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{1 + 3}}$

Step 10.4

Add $1$ and $3$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{4}}$

Step 10.5

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

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Step 10.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{m}$ as $m^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{(m^{\frac{1}{4}})}^{4}}$

Step 10.5.2

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

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{1}{4} \cdot 4}}$

Step 10.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{4}{4}}}$

Step 10.5.4

Cancel the common factor of $4$ .

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Step 10.5.4.1

Cancel the common factor.

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{4}{4}}}$

Step 10.5.4.2

Rewrite the expression.

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{1}}$

Step 10.5.5

Simplify.

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m}$

Step 11

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

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt{7} \sqrt[4]{m^{3}}}{m}$

Step 12

Simplify the numerator.

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Step 12.1

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

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Step 12.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{7^{\frac{1}{2}} \sqrt[4]{m^{3}}}{m}$

Step 12.1.2

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{7^{\frac{2}{4}} \sqrt[4]{m^{3}}}{m}$

Step 12.1.3

Rewrite $7^{\frac{2}{4}}$ as $\sqrt[4]{7^{2}}$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt[4]{7^{2}} \sqrt[4]{m^{3}}}{m}$

Step 12.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt[4]{7^{2} m^{3}}}{m}$

Step 12.3

Raise $7$ to the power of $2$ .

$\frac{\sqrt[4]{49 m^{3}}}{m} = \frac{\sqrt[4]{49 m^{3}}}{m}$

Step 13

Since the two sides have been shown to be equivalent, the equation is an identity.

$\sqrt[4]{\frac{7^{2}}{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$ is an identity.