Basic Math Examples

$3 + \frac{\sqrt{18}}{1 + \sqrt{8}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

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Step 1.1.1.1

Factor $9$ out of $18$ .

$3 + \frac{\sqrt{9 (2)}}{1 + \sqrt{8}}$

Step 1.1.1.2

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

$3 + \frac{\sqrt{3^{2} \cdot 2}}{1 + \sqrt{8}}$

Step 1.1.2

Pull terms out from under the radical.

$3 + \frac{3 \sqrt{2}}{1 + \sqrt{8}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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Step 1.2.1.1

Factor $4$ out of $8$ .

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

Step 1.2.1.2

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

$3 + \frac{3 \sqrt{2}}{1 + \sqrt{2^{2} \cdot 2}}$

Step 1.2.2

Pull terms out from under the radical.

$3 + \frac{3 \sqrt{2}}{1 + 2 \sqrt{2}}$

Step 1.3

Multiply $\frac{3 \sqrt{2}}{1 + 2 \sqrt{2}}$ by $\frac{1 - 2 \sqrt{2}}{1 - 2 \sqrt{2}}$ .

$3 + \frac{3 \sqrt{2}}{1 + 2 \sqrt{2}} \cdot \frac{1 - 2 \sqrt{2}}{1 - 2 \sqrt{2}}$

Step 1.4

Multiply $\frac{3 \sqrt{2}}{1 + 2 \sqrt{2}}$ by $\frac{1 - 2 \sqrt{2}}{1 - 2 \sqrt{2}}$ .

$3 + \frac{3 \sqrt{2} (1 - 2 \sqrt{2})}{(1 + 2 \sqrt{2}) (1 - 2 \sqrt{2})}$

Step 1.5

Expand the denominator using the FOIL method.

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

Step 1.6

Simplify.

$3 + \frac{3 \sqrt{2} (1 - 2 \sqrt{2})}{- 7}$

Step 1.7

Group $1 - 2 \sqrt{2}$ and $\sqrt{2}$ together.

$3 + \frac{3 ((1 - 2 \sqrt{2}) \sqrt{2})}{- 7}$

Step 1.8

Apply the distributive property.

$3 + \frac{3 (1 \sqrt{2} - 2 \sqrt{2} \sqrt{2})}{- 7}$

Step 1.9

Multiply $\sqrt{2}$ by $1$ .

$3 + \frac{3 (\sqrt{2} - 2 \sqrt{2} \sqrt{2})}{- 7}$

Step 1.10

Multiply $- 2 \sqrt{2} \sqrt{2}$ .

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Step 1.10.1

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

$3 + \frac{3 (\sqrt{2} - 2 ({\sqrt{2}}^{1} \sqrt{2}))}{- 7}$

Step 1.10.2

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

$3 + \frac{3 (\sqrt{2} - 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}))}{- 7}$

Step 1.10.3

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

$3 + \frac{3 (\sqrt{2} - 2 {\sqrt{2}}^{1 + 1})}{- 7}$

Step 1.10.4

Add $1$ and $1$ .

$3 + \frac{3 (\sqrt{2} - 2 {\sqrt{2}}^{2})}{- 7}$

Step 1.11

Simplify each term.

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Step 1.11.1

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

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Step 1.11.1.1

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

$3 + \frac{3 (\sqrt{2} - 2 {(2^{\frac{1}{2}})}^{2})}{- 7}$

Step 1.11.1.2

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

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

Step 1.11.1.3

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

$3 + \frac{3 (\sqrt{2} - 2 \cdot 2^{\frac{2}{2}})}{- 7}$

Step 1.11.1.4

Cancel the common factor of $2$ .

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Step 1.11.1.4.1

Cancel the common factor.

$3 + \frac{3 (\sqrt{2} - 2 \cdot 2^{\frac{2}{2}})}{- 7}$

Step 1.11.1.4.2

Rewrite the expression.

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

Step 1.11.1.5

Evaluate the exponent.

$3 + \frac{3 (\sqrt{2} - 2 \cdot 2)}{- 7}$

Step 1.11.2

Multiply $- 2$ by $2$ .

$3 + \frac{3 (\sqrt{2} - 4)}{- 7}$

Step 1.12

Move the negative in front of the fraction.

$3 - \frac{3 (\sqrt{2} - 4)}{7}$

Step 2

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$3 \cdot \frac{7}{7} - \frac{3 (\sqrt{2} - 4)}{7}$

Step 3

Combine fractions.

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Step 3.1

Combine $3$ and $\frac{7}{7}$ .

$\frac{3 \cdot 7}{7} - \frac{3 (\sqrt{2} - 4)}{7}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 \cdot 7 - 3 (\sqrt{2} - 4)}{7}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $3$ by $7$ .

$\frac{21 - 3 (\sqrt{2} - 4)}{7}$

Step 4.2

Apply the distributive property.

$\frac{21 - 3 \sqrt{2} - 3 \cdot - 4}{7}$

Step 4.3

Multiply $- 3$ by $- 4$ .

$\frac{21 - 3 \sqrt{2} + 12}{7}$

Step 4.4

Add $21$ and $12$ .

$\frac{33 - 3 \sqrt{2}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{33 - 3 \sqrt{2}}{7}$

Decimal Form:

$4.10819418 \dots$