Simplify (4x^7y)/( cube root of 27x^3y^5)

$\frac{4 x^{7} y}{\sqrt[3]{27 x^{3} y^{5}}}$

Step 1

Simplify the denominator.

Rewrite $27 x^{3} y^{5}$ as ${(3 x y)}^{3} y^{2}$ .

Rewrite $27$ as $3^{3}$ .

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

Factor out $y^{3}$ .

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

Rewrite $3^{3} x^{3} y^{3}$ as ${(3 x y)}^{3}$ .

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

Pull terms out from under the radical.

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

Step 2

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x^{7}$ and $x$ .

Factor $x$ out of $4 x^{7} y$ .

$\frac{x (4 x^{6} y)}{3 x y \sqrt[3]{y^{2}}}$

Cancel the common factors.

Factor $x$ out of $3 x y \sqrt[3]{y^{2}}$ .

$\frac{x (4 x^{6} y)}{x (3 y \sqrt[3]{y^{2}})}$

Cancel the common factor.

$\frac{x (4 x^{6} y)}{x (3 y \sqrt[3]{y^{2}})}$

Rewrite the expression.

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

Cancel the common factor of $y$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Multiply $\frac{4 x^{6}}{3 \sqrt[3]{y^{2}}}$ by $\frac{{\sqrt[3]{y^{2}}}^{2}}{{\sqrt[3]{y^{2}}}^{2}}$ .

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

Step 4

Combine and simplify the denominator.

Multiply $\frac{4 x^{6}}{3 \sqrt[3]{y^{2}}}$ by $\frac{{\sqrt[3]{y^{2}}}^{2}}{{\sqrt[3]{y^{2}}}^{2}}$ .

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

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

$\frac{4 x^{6} {\sqrt[3]{y^{2}}}^{2}}{3 (\sqrt[3]{y^{2}} {\sqrt[3]{y^{2}}}^{2})}$

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

$\frac{4 x^{6} {\sqrt[3]{y^{2}}}^{2}}{3 ({\sqrt[3]{y^{2}}}^{1} {\sqrt[3]{y^{2}}}^{2})}$

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

$\frac{4 x^{6} {\sqrt[3]{y^{2}}}^{2}}{3 {\sqrt[3]{y^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

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

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 $y^{\frac{2}{3}}$ .

$\frac{4 x^{6} {\sqrt[3]{y^{2}}}^{2}}{3 {(y^{\frac{2}{3}})}^{3}}$

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

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

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

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

Multiply $2$ by $3$ .

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

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

Factor $3$ out of $6$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

$\frac{4 x^{6} {\sqrt[3]{y^{2}}}^{2}}{3 y^{\frac{3 \cdot 2}{3 (1)}}}$

Cancel the common factor.

$\frac{4 x^{6} {\sqrt[3]{y^{2}}}^{2}}{3 y^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Rewrite the expression.

$\frac{4 x^{6} {\sqrt[3]{y^{2}}}^{2}}{3 y^{\frac{2}{1}}}$

Divide $2$ by $1$ .

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

Step 5

Simplify the numerator.

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

$\frac{4 x^{6} \sqrt[3]{{(y^{2})}^{2}}}{3 y^{2}}$

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

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

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

Multiply $2$ by $2$ .

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

Factor out $y^{3}$ .

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

Pull terms out from under the radical.

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

Step 6

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

Factor $y$ out of $4 x^{6} y \sqrt[3]{y}$ .

$\frac{y (4 x^{6} \sqrt[3]{y})}{3 y^{2}}$

Cancel the common factors.

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

$\frac{y (4 x^{6} \sqrt[3]{y})}{y (3 y)}$

Cancel the common factor.

$\frac{y (4 x^{6} \sqrt[3]{y})}{y (3 y)}$

Rewrite the expression.

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

Simplify (4x^7y)/( cube root of 27x^3y^5)

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