Simplify 3x^-2y^(1/2) cube root of 8x^4y

$3 x^{- 2} y^{\frac{1}{2}} \sqrt[3]{8 x^{4} y}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 \frac{1}{x^{2}} y^{\frac{1}{2}} \sqrt[3]{8 x^{4} y}$

Step 2

Combine $3$ and $\frac{1}{x^{2}}$ .

$\frac{3}{x^{2}} y^{\frac{1}{2}} \sqrt[3]{8 x^{4} y}$

Step 3

Combine $\frac{3}{x^{2}}$ and $y^{\frac{1}{2}}$ .

$\frac{3 y^{\frac{1}{2}}}{x^{2}} \sqrt[3]{8 x^{4} y}$

Step 4

Rewrite $8 x^{4} y$ as ${(2 x)}^{3} (x y)$ .

Rewrite $8$ as $2^{3}$ .

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

Factor out $x^{3}$ .

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

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

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

Multiply $2 \frac{3 y^{\frac{1}{2}}}{x^{2}}$ .

Combine $2$ and $\frac{3 y^{\frac{1}{2}}}{x^{2}}$ .

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

Multiply $3$ by $2$ .

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

Step 8

Cancel the common factor of $x$ .

Factor $x$ out of $x^{2}$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

Combine $\frac{6 y^{\frac{1}{2}}}{x}$ and $\sqrt[3]{x y}$ .

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