Algebra Examples

$3 \sqrt{2} \cdot 2 \sqrt{8} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 1

Multiply $2$ by $3$ .

$6 \sqrt{2} \sqrt{8} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 2

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1

Factor $4$ out of $8$ .

$6 \sqrt{2} \sqrt{4 (2)} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 2.2

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

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

Step 3

Pull terms out from under the radical.

$6 \sqrt{2} (| 2 | \sqrt{2}) \cdot \sqrt{3} \cdot \sqrt{6}$

Step 4

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 5

Multiply $6 \sqrt{2} (2 \sqrt{2})$ .

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Step 5.1

Multiply $2$ by $6$ .

$12 \sqrt{2} \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 5.2

Raise $\sqrt{2}$ to the power of $1$ .

$12 ({\sqrt{2}}^{1} \sqrt{2}) \cdot \sqrt{3} \cdot \sqrt{6}$

Step 5.3

Raise $\sqrt{2}$ to the power of $1$ .

$12 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) \cdot \sqrt{3} \cdot \sqrt{6}$

Step 5.4

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

$12 {\sqrt{2}}^{1 + 1} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 5.5

Add $1$ and $1$ .

$12 {\sqrt{2}}^{2} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$12 {(2^{\frac{1}{2}})}^{2} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$12 \cdot 2^{\frac{1}{2} \cdot 2} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 6.3

Combine $\frac{1}{2}$ and $2$ .

$12 \cdot 2^{\frac{2}{2}} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$12 \cdot 2^{\frac{2}{2}} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 6.4.2

Rewrite the expression.

$12 \cdot 2^{1} \cdot \sqrt{3} \cdot \sqrt{6}$

Step 6.5

Evaluate the exponent.

$12 \cdot 2 \cdot \sqrt{3} \cdot \sqrt{6}$

Step 7

Multiply $12$ by $2$ .

$24 \cdot \sqrt{3} \cdot \sqrt{6}$

Step 8

Multiply $24 \sqrt{3} \sqrt{6}$ .

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Step 8.1

Combine using the product rule for radicals.

$24 \sqrt{6 \cdot 3}$

Step 8.2

Multiply $6$ by $3$ .

$24 \sqrt{18}$

Step 9

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 9.1

Factor $9$ out of $18$ .

$24 \sqrt{9 (2)}$

Step 9.2

Rewrite $9$ as $3^{2}$ .

$24 \sqrt{3^{2} \cdot 2}$

Step 10

Pull terms out from under the radical.

$24 (| 3 | \sqrt{2})$

Step 11

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$24 (3 \sqrt{2})$

Step 12

Multiply $3$ by $24$ .

$72 \sqrt{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$72 \sqrt{2}$

Decimal Form:

$101.82337649 \dots$