Simplify ( square root of 6x^3)/( square root of 3x)* square root of 8x^2

$\frac{\sqrt{6 x^{3}}}{\sqrt{3 x}} \cdot \sqrt{8 x^{2}}$

Step 1

Simplify the numerator.

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

Factor out $x^{2}$ .

$\frac{\sqrt{6 (x^{2} x)}}{\sqrt{3 x}} \cdot \sqrt{8 x^{2}}$

Reorder $6$ and $x^{2}$ .

$\frac{\sqrt{x^{2} \cdot 6 x}}{\sqrt{3 x}} \cdot \sqrt{8 x^{2}}$

Add parentheses.

$\frac{\sqrt{x^{2} \cdot (6 x)}}{\sqrt{3 x}} \cdot \sqrt{8 x^{2}}$

Pull terms out from under the radical.

$\frac{x \sqrt{6 x}}{\sqrt{3 x}} \cdot \sqrt{8 x^{2}}$

Step 2

Combine $\sqrt{6 x}$ and $\sqrt{3 x}$ into a single radical.

$x \sqrt{\frac{6 x}{3 x}} \cdot \sqrt{8 x^{2}}$

Step 3

Reduce the expression $\frac{6 x}{3 x}$ by cancelling the common factors.

Factor $3$ out of $6 x$ .

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

Factor $3$ out of $3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x \sqrt{\frac{2 x}{x}} \cdot \sqrt{8 x^{2}}$

Step 4

Cancel the common factor of $x$ .

Cancel the common factor.

$x \sqrt{\frac{2 x}{x}} \cdot \sqrt{8 x^{2}}$

Divide $2$ by $1$ .

$x \sqrt{2} \cdot \sqrt{8 x^{2}}$

Step 5

Rewrite $8 x^{2}$ as ${(2 x)}^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Move $2$ .

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

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

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

Step 6

Pull terms out from under the radical.

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

Step 7

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Step 8

Multiply $x^{2} \sqrt{2} (2 \sqrt{2})$ .

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

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

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

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

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

$x^{2} (2 {\sqrt{2}}^{1 + 1})$

Add $1$ and $1$ .

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

Step 9

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{2} (2 \cdot 2^{1})$

Evaluate the exponent.

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

Step 10

Simplify the expression.

Multiply $2$ by $2$ .

$x^{2} \cdot 4$

Move $4$ to the left of $x^{2}$ .

$4 x^{2}$

Simplify ( square root of 6x^3)/( square root of 3x)* square root of 8x^2

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