Algebra Examples

$\frac{\sqrt{8 x^{2}} \cdot \sqrt{12 x^{3}}}{\sqrt{4 x^{4}}} x^{2}$

Step 1

Simplify the numerator.

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Step 1.1

Combine using the product rule for radicals.

$\frac{\sqrt{8 x^{2} (12 x^{3})}}{\sqrt{4 x^{4}}} \cdot x^{2}$

Step 1.2

Multiply $12$ by $8$ .

$\frac{\sqrt{96 x^{2} x^{3}}}{\sqrt{4 x^{4}}} \cdot x^{2}$

Step 1.3

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.3.1

Move $x^{3}$ .

$\frac{\sqrt{96 (x^{3} x^{2})}}{\sqrt{4 x^{4}}} \cdot x^{2}$

Step 1.3.2

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

$\frac{\sqrt{96 x^{3 + 2}}}{\sqrt{4 x^{4}}} \cdot x^{2}$

Step 1.3.3

Add $3$ and $2$ .

$\frac{\sqrt{96 x^{5}}}{\sqrt{4 x^{4}}} \cdot x^{2}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\sqrt{96 x^{5}}}{\sqrt{{(2 x^{2})}^{2}}} \cdot x^{2}$

Step 2.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{96 x^{5}}}{2 x^{2}} \cdot x^{2}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $96 x^{5}$ as ${(4 x^{2})}^{2} \cdot (6 x)$ .

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Step 3.1.1

Factor $16$ out of $96$ .

$\frac{\sqrt{16 (6) x^{5}}}{2 x^{2}} \cdot x^{2}$

Step 3.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} \cdot 6 x^{5}}}{2 x^{2}} \cdot x^{2}$

Step 3.1.3

Factor out $x^{4}$ .

$\frac{\sqrt{4^{2} \cdot 6 (x^{4} x)}}{2 x^{2}} \cdot x^{2}$

Step 3.1.4

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

$\frac{\sqrt{4^{2} \cdot 6 ({(x^{2})}^{2} x)}}{2 x^{2}} \cdot x^{2}$

Step 3.1.5

Move $6$ .

$\frac{\sqrt{4^{2} {(x^{2})}^{2} \cdot 6 x}}{2 x^{2}} \cdot x^{2}$

Step 3.1.6

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

$\frac{\sqrt{{(4 x^{2})}^{2} \cdot 6 x}}{2 x^{2}} \cdot x^{2}$

Step 3.1.7

Add parentheses.

$\frac{\sqrt{{(4 x^{2})}^{2} \cdot (6 x)}}{2 x^{2}} \cdot x^{2}$

Step 3.2

Pull terms out from under the radical.

$\frac{4 x^{2} \sqrt{6 x}}{2 x^{2}} \cdot x^{2}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

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

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Step 4.1.1

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

$\frac{4 x^{2} \sqrt{6 x}}{x^{2} \cdot 2} \cdot x^{2}$

Step 4.1.2

Cancel the common factor.

$\frac{4 x^{2} \sqrt{6 x}}{x^{2} \cdot 2} \cdot x^{2}$

Step 4.1.3

Rewrite the expression.

$\frac{4 x^{2} \sqrt{6 x}}{2}$

Step 4.2

Cancel the common factor of $4$ and $2$ .

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Step 4.2.1

Factor $2$ out of $4 x^{2} \sqrt{6 x}$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.2

Cancel the common factor.

$\frac{2 (2 x^{2} \sqrt{6 x})}{2 \cdot 1}$

Step 4.2.2.3

Rewrite the expression.

$\frac{2 x^{2} \sqrt{6 x}}{1}$

Step 4.2.2.4

Divide $2 x^{2} \sqrt{6 x}$ by $1$ .

$2 x^{2} \sqrt{6 x}$