Simplify ( square root of 8x^2* square root of 12x^3)/( square root of 4x^4)x^2

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

Step 1

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $12$ by $8$ .

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

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ .

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

Step 2

Simplify the denominator.

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

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

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.

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

Factor $16$ out of $96$ .

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

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

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

Factor out $x^{4}$ .

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

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

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

Move $6$ .

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

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

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

Add parentheses.

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

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.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify ( square root of 8x^2* square root of 12x^3)/( square root of 4x^4)x^2

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