Pre-Algebra Examples

$\frac{\sqrt{56 x^{3 y^{2}}}}{\sqrt{112 x^{2} y^{3}}}$

Step 1

Combine $\sqrt{56 x^{3 y^{2}}}$ and $\sqrt{112 x^{2} y^{3}}$ into a single radical.

$\sqrt{\frac{56 x^{3 y^{2}}}{112 x^{2} y^{3}}}$

Step 2

Reduce the expression $\frac{56 x^{3 y^{2}}}{112 x^{2} y^{3}}$ by cancelling the common factors.

Tap for more steps...

Step 2.1

Factor $56$ out of $56 x^{3 y^{2}}$ .

$\sqrt{\frac{56 (x^{3 y^{2}})}{112 x^{2} y^{3}}}$

Step 2.2

Factor $56$ out of $112 x^{2} y^{3}$ .

$\sqrt{\frac{56 (x^{3 y^{2}})}{56 (2 x^{2} y^{3})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{56 x^{3 y^{2}}}{56 (2 x^{2} y^{3})}}$

Step 2.4

Rewrite the expression.

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

Step 3

Cancel the common factor of $x^{3 y^{2}}$ and $x^{2}$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite $\sqrt{\frac{x^{3 y^{2} - 2}}{2 y}}$ as $\frac{\sqrt{x^{3 y^{2} - 2}}}{\sqrt{2 y}}$ .

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

Step 7

Combine.

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

Step 8

Multiply $\sqrt{x^{3 y^{2} - 2}}$ by $1$ .

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

Step 9

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

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

Step 10

Combine and simplify the denominator.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Move $\sqrt{2 y}$ .

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

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

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

Step 10.6

Add $1$ and $1$ .

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

Step 10.7

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

Tap for more steps...

Step 10.7.1

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

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

Step 10.7.2

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

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

Step 10.7.3

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

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

Step 10.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.7.4.1

Cancel the common factor.

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

Step 10.7.4.2

Rewrite the expression.

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

Step 10.7.5

Simplify.

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

Step 11

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 11.1

Move $y$ .

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

Step 11.2

Multiply $y$ by $y$ .

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

Step 12

Combine using the product rule for radicals.

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

Step 13

Move $2$ to the left of $y^{2}$ .

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

Step 14

Reorder factors in $\frac{\sqrt{x^{3 y^{2} - 2} \cdot 2 y}}{2 y^{2}}$ .

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