Linear Algebra Examples

$\frac{8}{\sqrt[7]{x y^{6}}}$

Step 1

Multiply $\frac{8}{\sqrt[7]{x y^{6}}}$ by $\frac{{\sqrt[7]{x y^{6}}}^{6}}{{\sqrt[7]{x y^{6}}}^{6}}$ .

$\frac{8}{\sqrt[7]{x y^{6}}} \cdot \frac{{\sqrt[7]{x y^{6}}}^{6}}{{\sqrt[7]{x y^{6}}}^{6}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{8}{\sqrt[7]{x y^{6}}}$ by $\frac{{\sqrt[7]{x y^{6}}}^{6}}{{\sqrt[7]{x y^{6}}}^{6}}$ .

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{\sqrt[7]{x y^{6}} {\sqrt[7]{x y^{6}}}^{6}}$

Step 2.2

Raise $\sqrt[7]{x y^{6}}$ to the power of $1$ .

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{\sqrt[7]{x y^{6}}}^{1} {\sqrt[7]{x y^{6}}}^{6}}$

Step 2.3

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

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{\sqrt[7]{x y^{6}}}^{1 + 6}}$

Step 2.4

Add $1$ and $6$ .

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{\sqrt[7]{x y^{6}}}^{7}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{({(x y^{6})}^{\frac{1}{7}})}^{7}}$

Step 2.5.2

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

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{(x y^{6})}^{\frac{1}{7} \cdot 7}}$

Step 2.5.3

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

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{(x y^{6})}^{\frac{7}{7}}}$

Step 2.5.4

Cancel the common factor of $7$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{(x y^{6})}^{\frac{7}{7}}}$

Step 2.5.4.2

Rewrite the expression.

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{{(x y^{6})}^{1}}$

Step 2.5.5

Simplify.

$\frac{8 {\sqrt[7]{x y^{6}}}^{6}}{x y^{6}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{8 \sqrt[7]{{(x y^{6})}^{6}}}{x y^{6}}$

Step 3.2

Apply the product rule to $x y^{6}$ .

$\frac{8 \sqrt[7]{x^{6} {(y^{6})}^{6}}}{x y^{6}}$

Step 3.3

Multiply the exponents in ${(y^{6})}^{6}$ .

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Step 3.3.1

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

$\frac{8 \sqrt[7]{x^{6} y^{6 \cdot 6}}}{x y^{6}}$

Step 3.3.2

Multiply $6$ by $6$ .

$\frac{8 \sqrt[7]{x^{6} y^{36}}}{x y^{6}}$

Step 3.4

Rewrite $x^{6} y^{36}$ as ${(y^{5})}^{7} (x^{6} y)$ .

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Step 3.4.1

Factor out $y^{35}$ .

$\frac{8 \sqrt[7]{x^{6} (y^{35} y)}}{x y^{6}}$

Step 3.4.2

Rewrite $y^{35}$ as ${(y^{5})}^{7}$ .

$\frac{8 \sqrt[7]{x^{6} ({(y^{5})}^{7} y)}}{x y^{6}}$

Step 3.4.3

Reorder $x^{6}$ and ${(y^{5})}^{7}$ .

$\frac{8 \sqrt[7]{{(y^{5})}^{7} x^{6} y}}{x y^{6}}$

Step 3.4.4

Add parentheses.

$\frac{8 \sqrt[7]{{(y^{5})}^{7} (x^{6} y)}}{x y^{6}}$

Step 3.5

Pull terms out from under the radical.

$\frac{8 y^{5} \sqrt[7]{x^{6} y}}{x y^{6}}$

Step 4

Cancel the common factor of $y^{5}$ and $y^{6}$ .

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Step 4.1

Factor $y^{5}$ out of $8 y^{5} \sqrt[7]{x^{6} y}$ .

$\frac{y^{5} (8 \sqrt[7]{x^{6} y})}{x y^{6}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $y^{5}$ out of $x y^{6}$ .

$\frac{y^{5} (8 \sqrt[7]{x^{6} y})}{y^{5} (x y)}$

Step 4.2.2

Cancel the common factor.

$\frac{y^{5} (8 \sqrt[7]{x^{6} y})}{y^{5} (x y)}$

Step 4.2.3

Rewrite the expression.

$\frac{8 \sqrt[7]{x^{6} y}}{x y}$