Algebra Examples

$\frac{\sqrt{6 x^{8} y^{9}}}{\sqrt{5 x^{2} y^{4}}}$

Step 1

Combine $\sqrt{6 x^{8} y^{9}}$ and $\sqrt{5 x^{2} y^{4}}$ into a single radical.

$\sqrt{\frac{6 x^{8} y^{9}}{5 x^{2} y^{4}}}$

Step 2

Reduce the expression $\frac{6 x^{8} y^{9}}{5 x^{2} y^{4}}$ by cancelling the common factors.

Tap for more steps...

Step 2.1

Factor $x^{2}$ out of $6 x^{8} y^{9}$ .

$\sqrt{\frac{x^{2} (6 x^{6} y^{9})}{5 x^{2} y^{4}}}$

Step 2.2

Factor $x^{2}$ out of $5 x^{2} y^{4}$ .

$\sqrt{\frac{x^{2} (6 x^{6} y^{9})}{x^{2} (5 y^{4})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{x^{2} (6 x^{6} y^{9})}{x^{2} (5 y^{4})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{6 x^{6} y^{9}}{5 y^{4}}}$

Step 3

Cancel the common factor of $y^{9}$ and $y^{4}$ .

Tap for more steps...

Step 3.1

Factor $y^{4}$ out of $6 x^{6} y^{9}$ .

$\sqrt{\frac{y^{4} (6 x^{6} y^{5})}{5 y^{4}}}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $y^{4}$ out of $5 y^{4}$ .

$\sqrt{\frac{y^{4} (6 x^{6} y^{5})}{y^{4} \cdot 5}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{y^{4} (6 x^{6} y^{5})}{y^{4} \cdot 5}}$

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Rewrite $6 x^{6} y^{5}$ as ${(x^{3} y^{2})}^{2} \cdot (6 y)$ .

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

Factor out $y^{4}$ .

$\frac{\sqrt{6 {(x^{3})}^{2} (y^{4} y)}}{\sqrt{5}}$

Step 5.1.3

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

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

Step 5.1.4

Move $6$ .

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

Step 5.1.5

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

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

Step 5.1.6

Add parentheses.

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

Step 5.2

Pull terms out from under the radical.

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

Step 6

Multiply $\frac{x^{3} y^{2} \sqrt{6 y}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 7

Combine and simplify the denominator.

Tap for more steps...

Step 7.1

Multiply $\frac{x^{3} y^{2} \sqrt{6 y}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 7.2

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 7.3

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 7.4

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

Tap for more steps...

Step 7.6.1

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{5}}{5^{1}}$

Step 7.6.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Combine using the product rule for radicals.

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

Step 8.2

Multiply $5$ by $6$ .

$\frac{x^{3} y^{2} \sqrt{30 y}}{5}$