Finite Math Examples

$\sqrt{2} (\sqrt{8} + 4 \sqrt{3})$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

$\sqrt{2} (\sqrt{4 (2)} + 4 \sqrt{3})$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

$\sqrt{2} (2 \sqrt{2} + 4 \sqrt{3})$

Step 2

Apply the distributive property.

$\sqrt{2} (2 \sqrt{2}) + \sqrt{2} (4 \sqrt{3})$

Step 3

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

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Step 3.1

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

$2 ({\sqrt{2}}^{1} \sqrt{2}) + \sqrt{2} (4 \sqrt{3})$

Step 3.2

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

$2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + \sqrt{2} (4 \sqrt{3})$

Step 3.3

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

$2 {\sqrt{2}}^{1 + 1} + \sqrt{2} (4 \sqrt{3})$

Step 3.4

Add $1$ and $1$ .

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

Step 4

Multiply $\sqrt{2} (4 \sqrt{3})$ .

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Step 4.1

Combine using the product rule for radicals.

$2 {\sqrt{2}}^{2} + 4 \sqrt{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

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

Step 5

Simplify each term.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

$2 \cdot 2^{\frac{1}{2} \cdot 2} + 4 \sqrt{6}$

Step 5.1.3

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

$2 \cdot 2^{\frac{2}{2}} + 4 \sqrt{6}$

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

$2 \cdot 2^{\frac{2}{2}} + 4 \sqrt{6}$

Step 5.1.4.2

Rewrite the expression.

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

Step 5.1.5

Evaluate the exponent.

$2 \cdot 2 + 4 \sqrt{6}$

Step 5.2

Multiply $2$ by $2$ .

$4 + 4 \sqrt{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$4 + 4 \sqrt{6}$

Decimal Form:

$13.79795897 \dots$