Simplify ( square root of 3x^12y^10)/( square root of 5x^6y^3)

$\frac{\sqrt{3 x^{12} y^{10}}}{\sqrt{5 x^{6} y^{3}}}$

Step 1

Combine $\sqrt{3 x^{12} y^{10}}$ and $\sqrt{5 x^{6} y^{3}}$ into a single radical.

$\sqrt{\frac{3 x^{12} y^{10}}{5 x^{6} y^{3}}}$

Step 2

Reduce the expression $\frac{3 x^{12} y^{10}}{5 x^{6} y^{3}}$ by cancelling the common factors.

Factor $x^{6}$ out of $3 x^{12} y^{10}$ .

$\sqrt{\frac{x^{6} (3 x^{6} y^{10})}{5 x^{6} y^{3}}}$

Factor $x^{6}$ out of $5 x^{6} y^{3}$ .

$\sqrt{\frac{x^{6} (3 x^{6} y^{10})}{x^{6} (5 y^{3})}}$

Cancel the common factor.

$\sqrt{\frac{x^{6} (3 x^{6} y^{10})}{x^{6} (5 y^{3})}}$

Rewrite the expression.

$\sqrt{\frac{3 x^{6} y^{10}}{5 y^{3}}}$

Step 3

Cancel the common factor of $y^{10}$ and $y^{3}$ .

Factor $y^{3}$ out of $3 x^{6} y^{10}$ .

$\sqrt{\frac{y^{3} (3 x^{6} y^{7})}{5 y^{3}}}$

Cancel the common factors.

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

$\sqrt{\frac{y^{3} (3 x^{6} y^{7})}{y^{3} \cdot 5}}$

Cancel the common factor.

$\sqrt{\frac{y^{3} (3 x^{6} y^{7})}{y^{3} \cdot 5}}$

Rewrite the expression.

$\sqrt{\frac{3 x^{6} y^{7}}{5}}$

Step 4

Rewrite $\sqrt{\frac{3 x^{6} y^{7}}{5}}$ as $\frac{\sqrt{3 x^{6} y^{7}}}{\sqrt{5}}$ .

$\frac{\sqrt{3 x^{6} y^{7}}}{\sqrt{5}}$

Step 5

Simplify the numerator.

Rewrite $3 x^{6} y^{7}$ as ${(x^{3} y^{3})}^{2} \cdot (3 y)$ .

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$\frac{\sqrt{3 {(x^{3})}^{2} y^{7}}}{\sqrt{5}}$

Factor out $y^{6}$ .

$\frac{\sqrt{3 {(x^{3})}^{2} (y^{6} y)}}{\sqrt{5}}$

Rewrite $y^{6}$ as ${(y^{3})}^{2}$ .

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

Move $3$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 6

Multiply $\frac{x^{3} y^{3} \sqrt{3 y}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 7

Combine and simplify the denominator.

Multiply $\frac{x^{3} y^{3} \sqrt{3 y}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{x^{3} y^{3} \sqrt{3 y} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{x^{3} y^{3} \sqrt{3 y} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$\frac{x^{3} y^{3} \sqrt{3 y} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{x^{3} y^{3} \sqrt{3 y} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$\frac{x^{3} y^{3} \sqrt{3 y} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x^{3} y^{3} \sqrt{3 y} \sqrt{5}}{5^{1}}$

Evaluate the exponent.

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

Step 8

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

$\frac{x^{3} y^{3} \sqrt{15 y}}{5}$

Simplify ( square root of 3x^12y^10)/( square root of 5x^6y^3)

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