Algebra Examples

$\sqrt{12} \cdot \sqrt{6} \cdot \sqrt{x^{3}} \cdot \sqrt{x^{7}}$

Step 1

Combine using the product rule for radicals.

$\sqrt{12 \cdot 6} \cdot \sqrt{x^{3}} \cdot \sqrt{x^{7}}$

Step 2

Multiply $12$ by $6$ .

$\sqrt{72} \cdot \sqrt{x^{3}} \cdot \sqrt{x^{7}}$

Step 3

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 3.1

Factor $36$ out of $72$ .

$\sqrt{36 (2)} \cdot \sqrt{x^{3}} \cdot \sqrt{x^{7}}$

Step 3.2

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2} \cdot 2} \cdot \sqrt{x^{3}} \cdot \sqrt{x^{7}}$

Step 4

Pull terms out from under the radical.

$6 \sqrt{2} \cdot \sqrt{x^{3}} \cdot \sqrt{x^{7}}$

Step 5

Factor out $x^{2}$ .

$6 \sqrt{2} \cdot \sqrt{x^{2} x} \cdot \sqrt{x^{7}}$

Step 6

Pull terms out from under the radical.

$6 \sqrt{2} \cdot (x \sqrt{x}) \cdot \sqrt{x^{7}}$

Step 7

Combine using the product rule for radicals.

$6 (x \sqrt{2 x}) \cdot \sqrt{x^{7}}$

Step 8

Rewrite $x^{7}$ as ${(x^{3})}^{2} x$ .

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Step 8.1

Factor out $x^{6}$ .

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

Step 8.2

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

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

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Step 10.1

Move $x^{3}$ .

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

Step 10.2

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

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Step 10.2.1

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

$6 (x^{3} x^{1}) \sqrt{2 x} \cdot \sqrt{x}$

Step 10.2.2

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

$6 x^{3 + 1} \sqrt{2 x} \cdot \sqrt{x}$

Step 10.3

Add $3$ and $1$ .

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

Step 11

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

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Step 11.1

Combine using the product rule for radicals.

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

Step 11.2

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

$6 x^{4} \sqrt{2 (x^{1} x)}$

Step 11.3

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

$6 x^{4} \sqrt{2 (x^{1} x^{1})}$

Step 11.4

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

$6 x^{4} \sqrt{2 x^{1 + 1}}$

Step 11.5

Add $1$ and $1$ .

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

Step 12

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

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

Step 13

Pull terms out from under the radical.

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

Step 14

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 14.1

Move $x$ .

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

Step 14.2

Multiply $x$ by $x^{4}$ .

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Step 14.2.1

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

$6 (x^{1} x^{4}) \sqrt{2}$

Step 14.2.2

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

$6 x^{1 + 4} \sqrt{2}$

Step 14.3

Add $1$ and $4$ .

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