Simplify ( square root of 45x^6y)/(3 square root of 15xy^3)

$\frac{\sqrt{45 x^{6} y}}{3 \sqrt{15 x y^{3}}}$

Step 1

Combine $\sqrt{45 x^{6} y}$ and $\sqrt{15 x y^{3}}$ into a single radical.

$\frac{\sqrt{\frac{45 x^{6} y}{15 x y^{3}}}}{3}$

Step 2

Reduce the expression by cancelling the common factors.

Cancel the common factor of $45$ and $15$ .

Factor $15$ out of $45 x^{6} y$ .

$\frac{\sqrt{\frac{15 (3 x^{6} y)}{15 x y^{3}}}}{3}$

Cancel the common factors.

Factor $15$ out of $15 x y^{3}$ .

$\frac{\sqrt{\frac{15 (3 x^{6} y)}{15 (x y^{3})}}}{3}$

Cancel the common factor.

$\frac{\sqrt{\frac{15 (3 x^{6} y)}{15 (x y^{3})}}}{3}$

Rewrite the expression.

$\frac{\sqrt{\frac{3 x^{6} y}{x y^{3}}}}{3}$

Cancel the common factor of $x^{6}$ and $x$ .

Factor $x$ out of $3 x^{6} y$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{\frac{3 x^{5} y}{y^{3}}}}{3}$

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

Factor $y$ out of $3 x^{5} y$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Simplify the numerator.

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

Factor the perfect power ${(x^{2})}^{2}$ out of $3 x^{5}$ .

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

Factor the perfect power $y^{2}$ out of $y^{2}$ .

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

Rearrange the fraction $\frac{{(x^{2})}^{2} \cdot (3 x)}{y^{2} \cdot 1}$ .

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

Pull terms out from under the radical.

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

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Simplify the expression.

Multiply $x^{2}$ by $1$ .

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

Move $3$ to the left of $y$ .

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

Simplify ( square root of 45x^6y)/(3 square root of 15xy^3)

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