Simplify ( seventh root of x^2)/( fifth root of y^3)

$\frac{\sqrt[7]{x^{2}}}{\sqrt[5]{y^{3}}}$

Step 1

Multiply $\frac{\sqrt[7]{x^{2}}}{\sqrt[5]{y^{3}}}$ by $\frac{{\sqrt[5]{y^{3}}}^{4}}{{\sqrt[5]{y^{3}}}^{4}}$ .

$\frac{\sqrt[7]{x^{2}}}{\sqrt[5]{y^{3}}} \cdot \frac{{\sqrt[5]{y^{3}}}^{4}}{{\sqrt[5]{y^{3}}}^{4}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[7]{x^{2}}}{\sqrt[5]{y^{3}}}$ by $\frac{{\sqrt[5]{y^{3}}}^{4}}{{\sqrt[5]{y^{3}}}^{4}}$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{\sqrt[5]{y^{3}} {\sqrt[5]{y^{3}}}^{4}}$

Raise $\sqrt[5]{y^{3}}$ to the power of $1$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{{\sqrt[5]{y^{3}}}^{1} {\sqrt[5]{y^{3}}}^{4}}$

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

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{{\sqrt[5]{y^{3}}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{{\sqrt[5]{y^{3}}}^{5}}$

Rewrite ${\sqrt[5]{y^{3}}}^{5}$ as $y^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{3}}$ as $y^{\frac{3}{5}}$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{{(y^{\frac{3}{5}})}^{5}}$

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

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{\frac{3}{5} \cdot 5}}$

Combine $\frac{3}{5}$ and $5$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{\frac{3 \cdot 5}{5}}}$

Multiply $3$ by $5$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{\frac{15}{5}}}$

Cancel the common factor of $15$ and $5$ .

Factor $5$ out of $15$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{\frac{5 \cdot 3}{5}}}$

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{\frac{5 \cdot 3}{5 (1)}}}$

Cancel the common factor.

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{\frac{5 \cdot 3}{5 \cdot 1}}}$

Rewrite the expression.

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{\frac{3}{1}}}$

Divide $3$ by $1$ .

$\frac{\sqrt[7]{x^{2}} {\sqrt[5]{y^{3}}}^{4}}{y^{3}}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[5]{y^{3}}}^{4}$ as ${({(y^{3})}^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[7]{x^{2}} \sqrt[5]{{(y^{3})}^{4}}}{y^{3}}$

Multiply the exponents in ${(y^{3})}^{4}$ .

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

$\frac{\sqrt[7]{x^{2}} \sqrt[5]{y^{3 \cdot 4}}}{y^{3}}$

Multiply $3$ by $4$ .

$\frac{\sqrt[7]{x^{2}} \sqrt[5]{y^{12}}}{y^{3}}$

Rewrite $y^{12}$ as ${(y^{2})}^{5} y^{2}$ .

Factor out $y^{10}$ .

$\frac{\sqrt[7]{x^{2}} \sqrt[5]{y^{10} y^{2}}}{y^{3}}$

Rewrite $y^{10}$ as ${(y^{2})}^{5}$ .

$\frac{\sqrt[7]{x^{2}} \sqrt[5]{{(y^{2})}^{5} y^{2}}}{y^{3}}$

Pull terms out from under the radical.

$\frac{\sqrt[7]{x^{2}} y^{2} \sqrt[5]{y^{2}}}{y^{3}}$

Combine exponents.

Rewrite the expression using the least common index of $35$ .

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

$\frac{y^{2} (x^{\frac{2}{7}} \sqrt[5]{y^{2}})}{y^{3}}$

Rewrite $x^{\frac{2}{7}}$ as $x^{\frac{10}{35}}$ .

$\frac{y^{2} (x^{\frac{10}{35}} \sqrt[5]{y^{2}})}{y^{3}}$

Rewrite $x^{\frac{10}{35}}$ as $\sqrt[35]{x^{10}}$ .

$\frac{y^{2} (\sqrt[35]{x^{10}} \sqrt[5]{y^{2}})}{y^{3}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{2}}$ as $y^{\frac{2}{5}}$ .

$\frac{y^{2} (\sqrt[35]{x^{10}} y^{\frac{2}{5}})}{y^{3}}$

Rewrite $y^{\frac{2}{5}}$ as $y^{\frac{14}{35}}$ .

$\frac{y^{2} (\sqrt[35]{x^{10}} y^{\frac{14}{35}})}{y^{3}}$

Rewrite $y^{\frac{14}{35}}$ as $\sqrt[35]{y^{14}}$ .

$\frac{y^{2} (\sqrt[35]{x^{10}} \sqrt[35]{y^{14}})}{y^{3}}$

Combine using the product rule for radicals.

$\frac{y^{2} \sqrt[35]{x^{10} y^{14}}}{y^{3}}$

Step 4

Cancel the common factor of $y^{2}$ and $y^{3}$ .

Factor $y^{2}$ out of $y^{2} \sqrt[35]{x^{10} y^{14}}$ .

$\frac{y^{2} (\sqrt[35]{x^{10} y^{14}})}{y^{3}}$

Cancel the common factors.

Factor $y^{2}$ out of $y^{3}$ .

$\frac{y^{2} (\sqrt[35]{x^{10} y^{14}})}{y^{2} y}$

Cancel the common factor.

$\frac{y^{2} \sqrt[35]{x^{10} y^{14}}}{y^{2} y}$

Rewrite the expression.

$\frac{\sqrt[35]{x^{10} y^{14}}}{y}$

Simplify ( seventh root of x^2)/( fifth root of y^3)

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