Simplify the Radical Expression ( fourth root of v^6)/( seventh root of v^5)

$\frac{\sqrt[4]{v^{6}}}{\sqrt[7]{v^{5}}}$

Step 1

Simplify the numerator.

Factor out $v^{4}$ .

$\frac{\sqrt[4]{v^{4} v^{2}}}{\sqrt[7]{v^{5}}}$

Pull terms out from under the radical.

$\frac{| v | \sqrt[4]{v^{2}}}{\sqrt[7]{v^{5}}}$

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

$\frac{| v | \sqrt{\sqrt{v^{2}}}}{\sqrt[7]{v^{5}}}$

Pull terms out from under the radical.

$\frac{| v | \sqrt{| v |}}{\sqrt[7]{v^{5}}}$

Step 2

Multiply $\frac{| v | \sqrt{| v |}}{\sqrt[7]{v^{5}}}$ by $\frac{{\sqrt[7]{v^{5}}}^{6}}{{\sqrt[7]{v^{5}}}^{6}}$ .

$\frac{| v | \sqrt{| v |}}{\sqrt[7]{v^{5}}} \cdot \frac{{\sqrt[7]{v^{5}}}^{6}}{{\sqrt[7]{v^{5}}}^{6}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{| v | \sqrt{| v |}}{\sqrt[7]{v^{5}}}$ by $\frac{{\sqrt[7]{v^{5}}}^{6}}{{\sqrt[7]{v^{5}}}^{6}}$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{\sqrt[7]{v^{5}} {\sqrt[7]{v^{5}}}^{6}}$

Raise $\sqrt[7]{v^{5}}$ to the power of $1$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{{\sqrt[7]{v^{5}}}^{1} {\sqrt[7]{v^{5}}}^{6}}$

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

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{{\sqrt[7]{v^{5}}}^{1 + 6}}$

Add $1$ and $6$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{{\sqrt[7]{v^{5}}}^{7}}$

Rewrite ${\sqrt[7]{v^{5}}}^{7}$ as $v^{5}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{v^{5}}$ as $v^{\frac{5}{7}}$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{{(v^{\frac{5}{7}})}^{7}}$

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

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{\frac{5}{7} \cdot 7}}$

Combine $\frac{5}{7}$ and $7$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{\frac{5 \cdot 7}{7}}}$

Multiply $5$ by $7$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{\frac{35}{7}}}$

Cancel the common factor of $35$ and $7$ .

Factor $7$ out of $35$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{\frac{7 \cdot 5}{7}}}$

Cancel the common factors.

Factor $7$ out of $7$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{\frac{7 \cdot 5}{7 (1)}}}$

Cancel the common factor.

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{\frac{7 \cdot 5}{7 \cdot 1}}}$

Rewrite the expression.

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{\frac{5}{1}}}$

Divide $5$ by $1$ .

$\frac{| v | \sqrt{| v |} {\sqrt[7]{v^{5}}}^{6}}{v^{5}}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[7]{v^{5}}}^{6}$ as ${({(v^{5})}^{6})}^{\frac{1}{7}}$ .

$\frac{| v | \sqrt{| v |} \sqrt[7]{{(v^{5})}^{6}}}{v^{5}}$

Multiply the exponents in ${(v^{5})}^{6}$ .

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

$\frac{| v | \sqrt{| v |} \sqrt[7]{v^{5 \cdot 6}}}{v^{5}}$

Multiply $5$ by $6$ .

$\frac{| v | \sqrt{| v |} \sqrt[7]{v^{30}}}{v^{5}}$

Rewrite $v^{30}$ as ${(v^{4})}^{7} v^{2}$ .

Factor out $v^{28}$ .

$\frac{| v | \sqrt{| v |} \sqrt[7]{v^{28} v^{2}}}{v^{5}}$

Rewrite $v^{28}$ as ${(v^{4})}^{7}$ .

$\frac{| v | \sqrt{| v |} \sqrt[7]{{(v^{4})}^{7} v^{2}}}{v^{5}}$

Pull terms out from under the radical.

$\frac{| v | \sqrt{| v |} v^{4} \sqrt[7]{v^{2}}}{v^{5}}$

Combine exponents.

Rewrite the expression using the least common index of $14$ .

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

$\frac{| v | v^{4} ({| v |}^{\frac{1}{2}} \sqrt[7]{v^{2}})}{v^{5}}$

Rewrite ${| v |}^{\frac{1}{2}}$ as ${| v |}^{\frac{7}{14}}$ .

$\frac{| v | v^{4} ({| v |}^{\frac{7}{14}} \sqrt[7]{v^{2}})}{v^{5}}$

Rewrite ${| v |}^{\frac{7}{14}}$ as $\sqrt[14]{{| v |}^{7}}$ .

$\frac{| v | v^{4} (\sqrt[14]{{| v |}^{7}} \sqrt[7]{v^{2}})}{v^{5}}$

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

$\frac{| v | v^{4} (\sqrt[14]{{| v |}^{7}} v^{\frac{2}{7}})}{v^{5}}$

Rewrite $v^{\frac{2}{7}}$ as $v^{\frac{4}{14}}$ .

$\frac{| v | v^{4} (\sqrt[14]{{| v |}^{7}} v^{\frac{4}{14}})}{v^{5}}$

Rewrite $v^{\frac{4}{14}}$ as $\sqrt[14]{v^{4}}$ .

$\frac{| v | v^{4} (\sqrt[14]{{| v |}^{7}} \sqrt[14]{v^{4}})}{v^{5}}$

Combine using the product rule for radicals.

$\frac{| v | v^{4} \sqrt[14]{{| v |}^{7} v^{4}}}{v^{5}}$

Step 5

Cancel the common factor of $v^{4}$ and $v^{5}$ .

Factor $v^{4}$ out of $| v | v^{4} \sqrt[14]{{| v |}^{7} v^{4}}$ .

$\frac{v^{4} (| v | \sqrt[14]{{| v |}^{7} v^{4}})}{v^{5}}$

Cancel the common factors.

Factor $v^{4}$ out of $v^{5}$ .

$\frac{v^{4} (| v | \sqrt[14]{{| v |}^{7} v^{4}})}{v^{4} v}$

Cancel the common factor.

$\frac{v^{4} (| v | \sqrt[14]{{| v |}^{7} v^{4}})}{v^{4} v}$

Rewrite the expression.

$\frac{| v | \sqrt[14]{{| v |}^{7} v^{4}}}{v}$

Step 6

Reorder factors in $\frac{| v | \sqrt[14]{{| v |}^{7} v^{4}}}{v}$ .

$\frac{| v | \sqrt[14]{v^{4} {| v |}^{7}}}{v}$

Simplify the Radical Expression ( fourth root of v^6)/( seventh root of v^5)

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