Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

$x^{3} \sqrt{3} \cdot \sqrt{8 x^{3}}$

Step 3

Rewrite $8 x^{3}$ as ${(2 x)}^{2} \cdot (2 x)$ .

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Step 3.1

Factor $4$ out of $8$ .

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

Step 3.2

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

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

Step 3.3

Factor out $x^{2}$ .

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

Step 3.4

Move $2$ .

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

Step 3.5

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

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

Step 3.6

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $x$ .

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

Step 5.2

Multiply $x$ by $x^{3}$ .

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Step 5.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{3} \sqrt{3} \cdot (2 \sqrt{2 x})$

Step 5.2.2

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

$x^{1 + 3} \sqrt{3} \cdot (2 \sqrt{2 x})$

Step 5.3

Add $1$ and $3$ .

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

Step 6

Multiply $x^{4} \sqrt{3} (2 \sqrt{2 x})$ .

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $3$ by $2$ .

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

Step 7

Rewrite using the commutative property of multiplication.

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