Algebra Examples

$\sqrt{7 x^{2}} \cdot \sqrt{12 x^{5}}$

Step 1

Reorder $7$ and $x^{2}$ .

$\sqrt{x^{2} \cdot 7} \cdot \sqrt{12 x^{5}}$

Step 2

Pull terms out from under the radical.

$x \sqrt{7} \cdot \sqrt{12 x^{5}}$

Step 3

Rewrite $12 x^{5}$ as ${(2 x^{2})}^{2} \cdot (3 x)$ .

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Step 3.1

Factor $4$ out of $12$ .

$x \sqrt{7} \cdot \sqrt{4 (3) x^{5}}$

Step 3.2

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

$x \sqrt{7} \cdot \sqrt{2^{2} \cdot 3 x^{5}}$

Step 3.3

Factor out $x^{4}$ .

$x \sqrt{7} \cdot \sqrt{2^{2} \cdot 3 (x^{4} x)}$

Step 3.4

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

$x \sqrt{7} \cdot \sqrt{2^{2} \cdot 3 ({(x^{2})}^{2} x)}$

Step 3.5

Move $3$ .

$x \sqrt{7} \cdot \sqrt{2^{2} {(x^{2})}^{2} \cdot 3 x}$

Step 3.6

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

$x \sqrt{7} \cdot \sqrt{{(2 x^{2})}^{2} \cdot 3 x}$

Step 3.7

Add parentheses.

$x \sqrt{7} \cdot \sqrt{{(2 x^{2})}^{2} \cdot (3 x)}$

Step 4

Pull terms out from under the radical.

$x \sqrt{7} \cdot (2 x^{2} \sqrt{3 x})$

Step 5

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

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Step 5.1

Move $x^{2}$ .

$x^{2} x \sqrt{7} \cdot (2 \sqrt{3 x})$

Step 5.2

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

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Step 5.2.1

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

$x^{2} x^{1} \sqrt{7} \cdot (2 \sqrt{3 x})$

Step 5.2.2

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

$x^{2 + 1} \sqrt{7} \cdot (2 \sqrt{3 x})$

Step 5.3

Add $2$ and $1$ .

$x^{3} \sqrt{7} \cdot (2 \sqrt{3 x})$

Step 6

Multiply $x^{3} \sqrt{7} (2 \sqrt{3 x})$ .

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Step 6.1

Combine using the product rule for radicals.

$x^{3} (2 \sqrt{7 (3 x)})$

Step 6.2

Multiply $7$ by $3$ .

$x^{3} (2 \sqrt{21 x})$

Step 7

Rewrite using the commutative property of multiplication.

$2 x^{3} \sqrt{21 x}$