Simplify square root of (64x^3y)/(2x^4)

$\sqrt{\frac{64 x^{3} y}{2 x^{4}}}$

Step 1

Reduce the expression $\frac{64 x^{3} y}{2 x^{4}}$ by cancelling the common factors.

Factor $2$ out of $64 x^{3} y$ .

$\sqrt{\frac{2 (32 x^{3} y)}{2 x^{4}}}$

Factor $2$ out of $2 x^{4}$ .

$\sqrt{\frac{2 (32 x^{3} y)}{2 (x^{4})}}$

Cancel the common factor.

$\sqrt{\frac{2 (32 x^{3} y)}{2 x^{4}}}$

Rewrite the expression.

$\sqrt{\frac{32 x^{3} y}{x^{4}}}$

Step 2

Cancel the common factor of $x^{3}$ and $x^{4}$ .

Factor $x^{3}$ out of $32 x^{3} y$ .

$\sqrt{\frac{x^{3} (32 y)}{x^{4}}}$

Cancel the common factors.

Factor $x^{3}$ out of $x^{4}$ .

$\sqrt{\frac{x^{3} (32 y)}{x^{3} x}}$

Cancel the common factor.

$\sqrt{\frac{x^{3} (32 y)}{x^{3} x}}$

Rewrite the expression.

$\sqrt{\frac{32 y}{x}}$

Step 3

Rewrite $\sqrt{\frac{32 y}{x}}$ as $\frac{\sqrt{32 y}}{\sqrt{x}}$ .

$\frac{\sqrt{32 y}}{\sqrt{x}}$

Step 4

Simplify the numerator.

Rewrite $32 y$ as ${(2^{2})}^{2} \cdot (2 y)$ .

Factor $16$ out of $32$ .

$\frac{\sqrt{16 (2) y}}{\sqrt{x}}$

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} \cdot 2 y}}{\sqrt{x}}$

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Raise $2$ to the power of $2$ .

$\frac{4 \sqrt{2 y}}{\sqrt{x}}$

Step 5

Multiply $\frac{4 \sqrt{2 y}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{4 \sqrt{2 y}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{4 \sqrt{2 y}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{4 \sqrt{2 y} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

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

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

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

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

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

$\frac{4 \sqrt{2 y} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{4 \sqrt{2 y} \sqrt{x}}{{\sqrt{x}}^{2}}$

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

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

$\frac{4 \sqrt{2 y} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

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

$\frac{4 \sqrt{2 y} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

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

$\frac{4 \sqrt{2 y} \sqrt{x}}{x^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 \sqrt{2 y} \sqrt{x}}{x^{\frac{2}{2}}}$

Rewrite the expression.

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

Simplify.

$\frac{4 \sqrt{2 y} \sqrt{x}}{x}$

Step 7

Combine using the product rule for radicals.

$\frac{4 \sqrt{x \cdot 2 y}}{x}$

Step 8

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

$\frac{4 \sqrt{2 x y}}{x}$

Simplify square root of (64x^3y)/(2x^4)

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