Simplify (4x^(4/3)y)/( cube root of 64x^5y^4)

$\frac{4 x^{\frac{4}{3}} y}{\sqrt[3]{64 x^{5} y^{4}}}$

Step 1

Simplify the denominator.

Rewrite $64 x^{5} y^{4}$ as ${(4 x y)}^{3} (x^{2} y)$ .

Rewrite $64$ as $4^{3}$ .

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

Factor out $x^{3}$ .

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

Factor out $y^{3}$ .

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

Move $x^{2}$ .

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

Rewrite $4^{3} (x^{3}) y^{3}$ as ${(4 x y)}^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 2

Move $x^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 3

Multiply $x^{\frac{4}{3}}$ by $x^{- 1}$ by adding the exponents.

Move $x^{- 1}$ .

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

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

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $3$ .

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

Add $- 3$ and $4$ .

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

Step 4

Cancel the common factor.

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

Step 5

Rewrite the expression.

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

Step 6

Cancel the common factor.

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

Step 7

Rewrite the expression.

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

Step 8

Multiply $\frac{x^{\frac{1}{3}}}{\sqrt[3]{x^{2} y}}$ by $\frac{{\sqrt[3]{x^{2} y}}^{2}}{{\sqrt[3]{x^{2} y}}^{2}}$ .

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

Step 9

Combine and simplify the denominator.

Multiply $\frac{x^{\frac{1}{3}}}{\sqrt[3]{x^{2} y}}$ by $\frac{{\sqrt[3]{x^{2} y}}^{2}}{{\sqrt[3]{x^{2} y}}^{2}}$ .

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

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

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

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

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

Add $1$ and $2$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} y}$ as ${(x^{2} y)}^{\frac{1}{3}}$ .

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 10

Move $x^{\frac{1}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 11

Multiply $x^{2}$ by $x^{- \frac{1}{3}}$ by adding the exponents.

Move $x^{- \frac{1}{3}}$ .

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

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

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{{\sqrt[3]{x^{2} y}}^{2}}{x^{- \frac{1}{3} + 2 \cdot \frac{3}{3}} y}$

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

$\frac{{\sqrt[3]{x^{2} y}}^{2}}{x^{- \frac{1}{3} + \frac{2 \cdot 3}{3}} y}$

Combine the numerators over the common denominator.

$\frac{{\sqrt[3]{x^{2} y}}^{2}}{x^{\frac{- 1 + 2 \cdot 3}{3}} y}$

Simplify the numerator.

Multiply $2$ by $3$ .

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

Add $- 1$ and $6$ .

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

Step 12

Simplify the numerator.

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

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

Apply the product rule to $x^{2} y$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Rewrite $x^{4} y^{2}$ as $x^{3} (x y^{2})$ .

Factor out $x^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 13

Move $x^{1}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 14

Multiply $x^{\frac{5}{3}}$ by $x^{- 1}$ by adding the exponents.

Move $x^{- 1}$ .

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

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

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $3$ .

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

Add $- 3$ and $5$ .

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

Step 15

Reorder factors in $\frac{\sqrt[3]{x y^{2}}}{x^{\frac{2}{3}} y}$ .

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

Simplify (4x^(4/3)y)/( cube root of 64x^5y^4)

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