Trigonometry Examples

$\frac{{(a - 6)}^{\frac{1}{7}}}{\sqrt[7]{a - 6}}$

Step 1

Multiply $\frac{{(a - 6)}^{\frac{1}{7}}}{\sqrt[7]{a - 6}}$ by $\frac{{\sqrt[7]{a - 6}}^{6}}{{\sqrt[7]{a - 6}}^{6}}$ .

$\frac{{(a - 6)}^{\frac{1}{7}}}{\sqrt[7]{a - 6}} \cdot \frac{{\sqrt[7]{a - 6}}^{6}}{{\sqrt[7]{a - 6}}^{6}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{{(a - 6)}^{\frac{1}{7}}}{\sqrt[7]{a - 6}}$ by $\frac{{\sqrt[7]{a - 6}}^{6}}{{\sqrt[7]{a - 6}}^{6}}$ .

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{\sqrt[7]{a - 6} {\sqrt[7]{a - 6}}^{6}}$

Step 2.2

Raise $\sqrt[7]{a - 6}$ to the power of $1$ .

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{\sqrt[7]{a - 6}}^{1} {\sqrt[7]{a - 6}}^{6}}$

Step 2.3

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

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{\sqrt[7]{a - 6}}^{1 + 6}}$

Step 2.4

Add $1$ and $6$ .

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{\sqrt[7]{a - 6}}^{7}}$

Step 2.5

Rewrite ${\sqrt[7]{a - 6}}^{7}$ as $a - 6$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{a - 6}$ as ${(a - 6)}^{\frac{1}{7}}$ .

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{({(a - 6)}^{\frac{1}{7}})}^{7}}$

Step 2.5.2

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

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{\frac{1}{7} \cdot 7}}$

Step 2.5.3

Combine $\frac{1}{7}$ and $7$ .

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{\frac{7}{7}}}$

Step 2.5.4

Cancel the common factor of $7$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{\frac{7}{7}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{1}}$

Step 2.5.5

Simplify.

$\frac{{(a - 6)}^{\frac{1}{7}} {\sqrt[7]{a - 6}}^{6}}{a - 6}$

Step 3

Move ${(a - 6)}^{\frac{1}{7}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{{\sqrt[7]{a - 6}}^{6}}{(a - 6) {(a - 6)}^{- \frac{1}{7}}}$

Step 4

Multiply $a - 6$ by ${(a - 6)}^{- \frac{1}{7}}$ by adding the exponents.

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Step 4.1

Multiply $a - 6$ by ${(a - 6)}^{- \frac{1}{7}}$ .

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Step 4.1.1

Raise $a - 6$ to the power of $1$ .

$\frac{{\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{1} {(a - 6)}^{- \frac{1}{7}}}$

Step 4.1.2

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

$\frac{{\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{1 - \frac{1}{7}}}$

Step 4.2

Write $1$ as a fraction with a common denominator.

$\frac{{\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{\frac{7}{7} - \frac{1}{7}}}$

Step 4.3

Combine the numerators over the common denominator.

$\frac{{\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{\frac{7 - 1}{7}}}$

Step 4.4

Subtract $1$ from $7$ .

$\frac{{\sqrt[7]{a - 6}}^{6}}{{(a - 6)}^{\frac{6}{7}}}$

Step 5

Rewrite ${\sqrt[7]{a - 6}}^{6}$ as $\sqrt[7]{{(a - 6)}^{6}}$ .

$\frac{\sqrt[7]{{(a - 6)}^{6}}}{{(a - 6)}^{\frac{6}{7}}}$