Algebra Examples

$\sqrt{5 x^{3}} \cdot \sqrt{12 x^{6}}$

Step 1

Rewrite $5 x^{3}$ as $x^{2} \cdot (5 x)$ .

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Step 1.1

Factor out $x^{2}$ .

$\sqrt{5 (x^{2} x)} \cdot \sqrt{12 x^{6}}$

Step 1.2

Reorder $5$ and $x^{2}$ .

$\sqrt{x^{2} \cdot 5 x} \cdot \sqrt{12 x^{6}}$

Step 1.3

Add parentheses.

$\sqrt{x^{2} \cdot (5 x)} \cdot \sqrt{12 x^{6}}$

Step 2

Pull terms out from under the radical.

$x \sqrt{5 x} \cdot \sqrt{12 x^{6}}$

Step 3

Rewrite $12 x^{6}$ as ${(2 x^{3})}^{2} \cdot 3$ .

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Step 3.1

Factor $4$ out of $12$ .

$x \sqrt{5 x} \cdot \sqrt{4 (3) x^{6}}$

Step 3.2

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

$x \sqrt{5 x} \cdot \sqrt{2^{2} \cdot 3 x^{6}}$

Step 3.3

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

$x \sqrt{5 x} \cdot \sqrt{2^{2} \cdot 3 {(x^{3})}^{2}}$

Step 3.4

Move $3$ .

$x \sqrt{5 x} \cdot \sqrt{2^{2} {(x^{3})}^{2} \cdot 3}$

Step 3.5

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

$x \sqrt{5 x} \cdot \sqrt{{(2 x^{3})}^{2} \cdot 3}$

Step 4

Pull terms out from under the radical.

$x \sqrt{5 x} \cdot (2 x^{3} \sqrt{3})$

Step 5

Rewrite using the commutative property of multiplication.

$2 \sqrt{3} x \sqrt{5 x} x^{3}$

Step 6

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

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Step 6.1

Move $x^{3}$ .

$2 \sqrt{3} (x^{3} x) \sqrt{5 x}$

Step 6.2

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

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Step 6.2.1

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

$2 \sqrt{3} (x^{3} x^{1}) \sqrt{5 x}$

Step 6.2.2

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

$2 \sqrt{3} x^{3 + 1} \sqrt{5 x}$

Step 6.3

Add $3$ and $1$ .

$2 \sqrt{3} x^{4} \sqrt{5 x}$

Step 7

Multiply $2 \sqrt{3} x^{4} \sqrt{5 x}$ .

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Step 7.1

Combine using the product rule for radicals.

$2 x^{4} \sqrt{3 (5 x)}$

Step 7.2

Multiply $3$ by $5$ .

$2 x^{4} \sqrt{15 x}$