Algebra Examples

$\frac{\sqrt{12 x^{6}}}{\sqrt{3 x^{4}}} \cdot \sqrt{4 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $12 x^{6}$ as ${(2 x^{3})}^{2} \cdot 3$ .

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Step 1.1.1

Factor $4$ out of $12$ .

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

Step 1.1.2

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

$\frac{\sqrt{2^{2} \cdot 3 x^{6}}}{\sqrt{3 x^{4}}} \cdot \sqrt{4 x^{2}}$

Step 1.1.3

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

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

Step 1.1.4

Move $3$ .

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

Step 1.1.5

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $3 x^{4}$ as ${(x^{2})}^{2} \cdot 3$ .

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Step 2.1.1

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

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

Step 2.1.2

Reorder $3$ and ${(x^{2})}^{2}$ .

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

Factor $x^{2}$ out of $2 x^{3} \sqrt{3}$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $x^{2}$ out of $x^{2} \sqrt{3}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $\sqrt{3}$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Divide $2 x$ by $1$ .

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

Step 3.3

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

$2 x \cdot \sqrt{{(2 x)}^{2}}$

Step 4

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

$2 x \cdot (2 x)$

Step 5

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x$

Step 6

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1

Move $x$ .

$2 \cdot 2 (x \cdot x)$

Step 6.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2}$

Step 7

Multiply $2$ by $2$ .

$4 x^{2}$