Simplify ( square root of 15xy)/(3 square root of 10xy^3)

$\frac{\sqrt{15 x y}}{3 \sqrt{10 x y^{3}}}$

Step 1

Combine $\sqrt{15 x y}$ and $\sqrt{10 x y^{3}}$ into a single radical.

$\frac{\sqrt{\frac{15 x y}{10 x y^{3}}}}{3}$

Step 2

Reduce the expression by cancelling the common factors.

Cancel the common factor of $15$ and $10$ .

Factor $5$ out of $15 x y$ .

$\frac{\sqrt{\frac{5 (3 x y)}{10 x y^{3}}}}{3}$

Cancel the common factors.

Factor $5$ out of $10 x y^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $y$ out of $3 y$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

Simplify the numerator.

Rewrite $\frac{3}{2 y^{2}}$ as ${(\frac{1}{y})}^{2} \frac{3}{2}$ .

Factor the perfect power $1^{2}$ out of $3$ .

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

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

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

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

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

Pull terms out from under the radical.

$\frac{\frac{1}{y} \sqrt{\frac{3}{2}}}{3}$

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

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

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

$\frac{\frac{1}{y} \cdot \frac{\sqrt{6}}{2}}{3}$

Multiply $\frac{1}{y}$ by $\frac{\sqrt{6}}{2}$ .

$\frac{\frac{\sqrt{6}}{y \cdot 2}}{3}$

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

$\frac{\frac{\sqrt{6}}{2 y}}{3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{6}}{2 y} \cdot \frac{1}{3}$

Step 5

Multiply $\frac{\sqrt{6}}{2 y} \cdot \frac{1}{3}$ .

Multiply $\frac{\sqrt{6}}{2 y}$ by $\frac{1}{3}$ .

$\frac{\sqrt{6}}{2 y \cdot 3}$

Multiply $3$ by $2$ .

$\frac{\sqrt{6}}{6 y}$

Simplify ( square root of 15xy)/(3 square root of 10xy^3)

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