Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Factor $4$ out of $8$ .

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

Step 3.2

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

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

Step 3.3

Factor out $x^{2}$ .

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

Step 3.4

Move $2$ .

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

Step 3.5

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

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

Step 3.6

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $x$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.3

Add $1$ and $2$ .

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

Step 6

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

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Multiply $2$ by $2$ .

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

Step 7

Simplify the expression.

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Step 7.1

Rewrite using the commutative property of multiplication.

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

Step 7.2

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $2$ by $2$ .

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