Pre-Algebra Examples

$\frac{\sqrt{6 x^{11} y^{12}}}{\sqrt{7 x^{5} y^{7}}}$

Step 1

Combine $\sqrt{6 x^{11} y^{12}}$ and $\sqrt{7 x^{5} y^{7}}$ into a single radical.

$\sqrt{\frac{6 x^{11} y^{12}}{7 x^{5} y^{7}}}$

Step 2

Reduce the expression $\frac{6 x^{11} y^{12}}{7 x^{5} y^{7}}$ by cancelling the common factors.

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Step 2.1

Factor $x^{5}$ out of $6 x^{11} y^{12}$ .

$\sqrt{\frac{x^{5} (6 x^{6} y^{12})}{7 x^{5} y^{7}}}$

Step 2.2

Factor $x^{5}$ out of $7 x^{5} y^{7}$ .

$\sqrt{\frac{x^{5} (6 x^{6} y^{12})}{x^{5} (7 y^{7})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{x^{5} (6 x^{6} y^{12})}{x^{5} (7 y^{7})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{6 x^{6} y^{12}}{7 y^{7}}}$

Step 3

Cancel the common factor of $y^{12}$ and $y^{7}$ .

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Step 3.1

Factor $y^{7}$ out of $6 x^{6} y^{12}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $y^{7}$ out of $7 y^{7}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

$\frac{\sqrt{6 {(x^{3})}^{2} y^{5}}}{\sqrt{7}}$

Step 5.1.2

Factor out $y^{4}$ .

$\frac{\sqrt{6 {(x^{3})}^{2} (y^{4} y)}}{\sqrt{7}}$

Step 5.1.3

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

$\frac{\sqrt{6 {(x^{3})}^{2} ({(y^{2})}^{2} y)}}{\sqrt{7}}$

Step 5.1.4

Move $6$ .

$\frac{\sqrt{{(x^{3})}^{2} {(y^{2})}^{2} \cdot 6 y}}{\sqrt{7}}$

Step 5.1.5

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

$\frac{\sqrt{{(x^{3} y^{2})}^{2} \cdot 6 y}}{\sqrt{7}}$

Step 5.1.6

Add parentheses.

$\frac{\sqrt{{(x^{3} y^{2})}^{2} \cdot (6 y)}}{\sqrt{7}}$

Step 5.2

Pull terms out from under the radical.

$\frac{x^{3} y^{2} \sqrt{6 y}}{\sqrt{7}}$

Step 6

Multiply $\frac{x^{3} y^{2} \sqrt{6 y}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{x^{3} y^{2} \sqrt{6 y}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{x^{3} y^{2} \sqrt{6 y}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 7.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 7.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 7.4

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 7.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{7^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{x^{3} y^{2} \sqrt{6 y} \sqrt{7}}{7}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$\frac{x^{3} y^{2} \sqrt{7 (6 y)}}{7}$

Step 8.2

Multiply $7$ by $6$ .

$\frac{x^{3} y^{2} \sqrt{42 y}}{7}$