Simplify ( fourth root of y* square root of y)/( cube root of y)

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

Step 1

Simplify the numerator.

Rewrite the expression using the least common index of $4$ .

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

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

Rewrite $y^{\frac{1}{2}}$ as $y^{\frac{2}{4}}$ .

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

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

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

Combine using the product rule for radicals.

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

Multiply $y$ by $y^{2}$ by adding the exponents.

Multiply $y$ by $y^{2}$ .

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

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

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

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

Add $1$ and $2$ .

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Rewrite ${\sqrt[3]{y}}^{2}$ as ${(y^{2})}^{\frac{1}{3}}$ .

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

Step 5

Simplify the numerator.

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y^{3}}$ as $y^{\frac{3}{4}}$ .

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

Rewrite $y^{\frac{3}{4}}$ as $y^{\frac{9}{12}}$ .

$\frac{y^{\frac{9}{12}} \sqrt[3]{y^{2}}}{y}$

Rewrite $y^{\frac{9}{12}}$ as $\sqrt[12]{y^{9}}$ .

$\frac{\sqrt[12]{y^{9}} \sqrt[3]{y^{2}}}{y}$

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

$\frac{\sqrt[12]{y^{9}} y^{\frac{2}{3}}}{y}$

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{8}{12}}$ .

$\frac{\sqrt[12]{y^{9}} y^{\frac{8}{12}}}{y}$

Rewrite $y^{\frac{8}{12}}$ as $\sqrt[12]{y^{8}}$ .

$\frac{\sqrt[12]{y^{9}} \sqrt[12]{y^{8}}}{y}$

Combine using the product rule for radicals.

$\frac{\sqrt[12]{y^{9} y^{8}}}{y}$

Multiply $y^{9}$ by $y^{8}$ by adding the exponents.

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

$\frac{\sqrt[12]{y^{9 + 8}}}{y}$

Add $9$ and $8$ .

$\frac{\sqrt[12]{y^{17}}}{y}$

Step 6

Simplify the numerator.

Factor out $y^{12}$ .

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

Pull terms out from under the radical.

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

Step 7

Cancel the common factor of $y$ .

Cancel the common factor.

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

Divide $\sqrt[12]{y^{5}}$ by $1$ .

$\sqrt[12]{y^{5}}$

Simplify ( fourth root of y* square root of y)/( cube root of y)

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