Simplify ( fifth root of y^4)/( sixth root of y^4)

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

Step 1

Simplify the denominator.

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

$\frac{\sqrt[5]{y^{4}}}{\sqrt[6]{{(y^{2})}^{2}}}$

Rewrite $\sqrt[6]{{(y^{2})}^{2}}$ as $\sqrt[3]{\sqrt{{(y^{2})}^{2}}}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{y^{2}}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{y^{2}}}^{3}$ as $y^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{2}}$ as $y^{\frac{2}{3}}$ .

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

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

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

Combine $\frac{2}{3}$ and $3$ .

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

Multiply $2$ by $3$ .

$\frac{\sqrt[5]{y^{4}} {\sqrt[3]{y^{2}}}^{2}}{y^{\frac{6}{3}}}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

Step 4

Simplify the numerator.

Rewrite ${\sqrt[3]{y^{2}}}^{2}$ as $\sqrt[3]{{(y^{2})}^{2}}$ .

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

Multiply the exponents in ${(y^{2})}^{2}$ .

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

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

Multiply $2$ by $2$ .

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

Factor out $y^{3}$ .

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

Pull terms out from under the radical.

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

Combine exponents.

Rewrite the expression using the least common index of $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{4}}$ as $y^{\frac{4}{5}}$ .

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

Rewrite $y^{\frac{4}{5}}$ as $y^{\frac{12}{15}}$ .

$\frac{y (y^{\frac{12}{15}} \sqrt[3]{y})}{y^{2}}$

Rewrite $y^{\frac{12}{15}}$ as $\sqrt[15]{y^{12}}$ .

$\frac{y (\sqrt[15]{y^{12}} \sqrt[3]{y})}{y^{2}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

$\frac{y (\sqrt[15]{y^{12}} y^{\frac{1}{3}})}{y^{2}}$

Rewrite $y^{\frac{1}{3}}$ as $y^{\frac{5}{15}}$ .

$\frac{y (\sqrt[15]{y^{12}} y^{\frac{5}{15}})}{y^{2}}$

Rewrite $y^{\frac{5}{15}}$ as $\sqrt[15]{y^{5}}$ .

$\frac{y (\sqrt[15]{y^{12}} \sqrt[15]{y^{5}})}{y^{2}}$

Combine using the product rule for radicals.

$\frac{y \sqrt[15]{y^{12} y^{5}}}{y^{2}}$

Multiply $y^{12}$ by $y^{5}$ by adding the exponents.

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

$\frac{y \sqrt[15]{y^{12 + 5}}}{y^{2}}$

Add $12$ and $5$ .

$\frac{y \sqrt[15]{y^{17}}}{y^{2}}$

Factor out $y^{15}$ .

$\frac{y \sqrt[15]{y^{15} y^{2}}}{y^{2}}$

Pull terms out from under the radical.

$\frac{y (y \sqrt[15]{y^{2}})}{y^{2}}$

Step 5

Simplify the numerator.

Raise $y$ to the power of $1$ .

$\frac{y^{1} y \sqrt[15]{y^{2}}}{y^{2}}$

Raise $y$ to the power of $1$ .

$\frac{y^{1} y^{1} \sqrt[15]{y^{2}}}{y^{2}}$

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

$\frac{y^{1 + 1} \sqrt[15]{y^{2}}}{y^{2}}$

Add $1$ and $1$ .

$\frac{y^{2} \sqrt[15]{y^{2}}}{y^{2}}$

Step 6

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

Cancel the common factor.

$\frac{y^{2} \sqrt[15]{y^{2}}}{y^{2}}$

Divide $\sqrt[15]{y^{2}}$ by $1$ .

$\sqrt[15]{y^{2}}$

Simplify ( fifth root of y^4)/( sixth root of y^4)

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