Trigonometry Examples

$\sqrt[3]{\frac{6}{7}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 1

Rewrite $\sqrt[3]{\frac{6}{7}}$ as $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ .

$\frac{\sqrt[3]{6}}{\sqrt[3]{7}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 2

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$\frac{\sqrt[3]{6}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.3

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

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.5

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.5.2

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

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{1}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 3.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$\frac{\sqrt[3]{6} \sqrt[3]{7^{2}}}{7} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 4.2

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

$\frac{\sqrt[3]{6} \sqrt[3]{49}}{7} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{6 \cdot 49}}{7} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 5.2

Multiply $6$ by $49$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$

Step 6

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Step 7.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}}$

Step 7.3

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

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Step 7.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Step 7.5

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

Step 7.5.2

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

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

Step 7.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 7.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7^{1}}$

Step 7.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} {\sqrt[3]{7}}^{2}}{7}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} \sqrt[3]{7^{2}}}{7}$

Step 8.2

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

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6} \sqrt[3]{49}}{7}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{6 \cdot 49}}{7}$

Step 9.2

Multiply $6$ by $49$ .

$\frac{\sqrt[3]{294}}{7} = \frac{\sqrt[3]{294}}{7}$

Step 10

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

$\sqrt[3]{\frac{6}{7}} = \frac{\sqrt[3]{6}}{\sqrt[3]{7}}$ is an identity.