Calculus Examples

$\frac{1}{\sqrt[7]{x}}$

Step 1

Multiply $\frac{1}{\sqrt[7]{x}}$ by $\frac{{\sqrt[7]{x}}^{6}}{{\sqrt[7]{x}}^{6}}$ .

$\frac{1}{\sqrt[7]{x}} \cdot \frac{{\sqrt[7]{x}}^{6}}{{\sqrt[7]{x}}^{6}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1}{\sqrt[7]{x}}$ by $\frac{{\sqrt[7]{x}}^{6}}{{\sqrt[7]{x}}^{6}}$ .

$\frac{{\sqrt[7]{x}}^{6}}{\sqrt[7]{x} {\sqrt[7]{x}}^{6}}$

Step 2.2

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

$\frac{{\sqrt[7]{x}}^{6}}{{\sqrt[7]{x}}^{1} {\sqrt[7]{x}}^{6}}$

Step 2.3

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

$\frac{{\sqrt[7]{x}}^{6}}{{\sqrt[7]{x}}^{1 + 6}}$

Step 2.4

Add $1$ and $6$ .

$\frac{{\sqrt[7]{x}}^{6}}{{\sqrt[7]{x}}^{7}}$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

$\frac{{\sqrt[7]{x}}^{6}}{x^{\frac{1}{7} \cdot 7}}$

Step 2.5.3

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

$\frac{{\sqrt[7]{x}}^{6}}{x^{\frac{7}{7}}}$

Step 2.5.4

Cancel the common factor of $7$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{{\sqrt[7]{x}}^{6}}{x^{\frac{7}{7}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{{\sqrt[7]{x}}^{6}}{x^{1}}$

Step 2.5.5

Simplify.

$\frac{{\sqrt[7]{x}}^{6}}{x}$

Step 3

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

$\frac{\sqrt[7]{x^{6}}}{x}$