Algebra Examples

$\sqrt{6 x^{5}} \cdot \sqrt{8 x^{2}}$

Step 1

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

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Step 1.1

Factor out $x^{4}$ .

$\sqrt{6 (x^{4} x)} \cdot \sqrt{8 x^{2}}$

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Factor $4$ out of $8$ .

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

Step 3.2

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

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

Step 3.3

Move $2$ .

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

Step 3.4

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

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Step 6.1

Move $x$ .

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

Step 6.2

Multiply $x$ by $x^{2}$ .

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.3

Add $1$ and $2$ .

$2 \sqrt{2} x^{3} \sqrt{6 x}$

Step 7

Multiply $2 \sqrt{2} x^{3} \sqrt{6 x}$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $2$ by $6$ .

$2 x^{3} \sqrt{12 x}$

Step 8

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

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Step 8.1

Factor $4$ out of $12$ .

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

Step 8.2

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

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

Step 8.3

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

Multiply $2$ by $2$ .

$4 x^{3} \sqrt{3 x}$