Simplify square root of 8x^3* square root of 2xy^2

$\sqrt{8 x^{3}} \cdot \sqrt{2 x y^{2}}$

Step 1

Rewrite $8 x^{3}$ as ${(2 x)}^{2} \cdot (2 x)$ .

Factor $4$ out of $8$ .

$\sqrt{4 (2) x^{3}} \cdot \sqrt{2 x y^{2}}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 2 x^{3}} \cdot \sqrt{2 x y^{2}}$

Factor out $x^{2}$ .

$\sqrt{2^{2} \cdot 2 (x^{2} x)} \cdot \sqrt{2 x y^{2}}$

Move $2$ .

$\sqrt{2^{2} x^{2} \cdot 2 x} \cdot \sqrt{2 x y^{2}}$

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

$\sqrt{{(2 x)}^{2} \cdot 2 x} \cdot \sqrt{2 x y^{2}}$

Add parentheses.

$\sqrt{{(2 x)}^{2} \cdot (2 x)} \cdot \sqrt{2 x y^{2}}$

Step 2

Pull terms out from under the radical.

$2 x \sqrt{2 x} \cdot \sqrt{2 x y^{2}}$

Step 3

Rewrite $2 x y^{2}$ as $y^{2} \cdot (2 x)$ .

Move $x$ .

$2 x \sqrt{2 x} \cdot \sqrt{2 y^{2} x}$

Reorder $2$ and $y^{2}$ .

$2 x \sqrt{2 x} \cdot \sqrt{y^{2} \cdot 2 x}$

Add parentheses.

$2 x \sqrt{2 x} \cdot \sqrt{y^{2} \cdot (2 x)}$

Step 4

Pull terms out from under the radical.

$2 x \sqrt{2 x} \cdot (y \sqrt{2 x})$

Step 5

Multiply $2 x \sqrt{2 x} (y \sqrt{2 x})$ .

Raise $\sqrt{2 x}$ to the power of $1$ .

$2 x (y ({\sqrt{2 x}}^{1} \sqrt{2 x}))$

Raise $\sqrt{2 x}$ to the power of $1$ .

$2 x (y ({\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}))$

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

$2 x (y {\sqrt{2 x}}^{1 + 1})$

Add $1$ and $1$ .

$2 x (y {\sqrt{2 x}}^{2})$

Step 6

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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

$2 x (y {({(2 x)}^{\frac{1}{2}})}^{2})$

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

$2 x (y {(2 x)}^{\frac{1}{2} \cdot 2})$

Combine $\frac{1}{2}$ and $2$ .

$2 x (y {(2 x)}^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x (y {(2 x)}^{\frac{2}{2}})$

Rewrite the expression.

$2 x (y {(2 x)}^{1})$

Simplify.

$2 x (y (2 x))$

Step 7

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$2 (x \cdot x) (y \cdot (2))$

Multiply $x$ by $x$ .

$2 x^{2} (y \cdot (2))$

Step 8

Move $2$ to the left of $y$ .

$2 x^{2} (2 y)$

Step 9

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x^{2} y$

Step 10

Multiply $2$ by $2$ .

$4 x^{2} y$

Simplify square root of 8x^3* square root of 2xy^2

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