Basic Math Examples

$\frac{1}{3 \sqrt{2}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

$\frac{\sqrt{2}}{3 \sqrt{2} \sqrt{2}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.2

Move $\sqrt{2}$ .

$\frac{\sqrt{2}}{3 (\sqrt{2} \sqrt{2})} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.3

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

$\frac{\sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.4

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

$\frac{\sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.5

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

$\frac{\sqrt{2}}{3 {\sqrt{2}}^{1 + 1}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.6

Add $1$ and $1$ .

$\frac{\sqrt{2}}{3 {\sqrt{2}}^{2}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.7

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

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Step 1.2.7.1

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

$\frac{\sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.7.2

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

$\frac{\sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.7.3

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

$\frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.7.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{3 \cdot 2^{1}} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.2.7.5

Evaluate the exponent.

$\frac{\sqrt{2}}{3 \cdot 2} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.3

Multiply $3$ by $2$ .

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{1}{6 \sqrt{2}}$

Step 1.4

Multiply $\frac{1}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - (\frac{1}{6 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

Multiply $\frac{1}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 \sqrt{2} \sqrt{2}}$

Step 1.5.2

Move $\sqrt{2}$ .

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 1.5.3

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

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 1.5.4

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

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 1.5.5

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

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 {\sqrt{2}}^{1 + 1}}$

Step 1.5.6

Add $1$ and $1$ .

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 {\sqrt{2}}^{2}}$

Step 1.5.7

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

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Step 1.5.7.1

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

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 {(2^{\frac{1}{2}})}^{2}}$

Step 1.5.7.2

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

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 1.5.7.3

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

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 1.5.7.4

Cancel the common factor of $2$ .

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Step 1.5.7.4.1

Cancel the common factor.

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 1.5.7.4.2

Rewrite the expression.

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 \cdot 2^{1}}$

Step 1.5.7.5

Evaluate the exponent.

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{6 \cdot 2}$

Step 1.6

Multiply $6$ by $2$ .

$\frac{\sqrt{2}}{6} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{12}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{\sqrt{2}}{6}$ by $\frac{2}{2}$ .

$\frac{\sqrt{2}}{6} \cdot \frac{2}{2} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{12}$

Step 2.2

Multiply $\frac{\sqrt{2}}{6}$ by $\frac{2}{2}$ .

$\frac{\sqrt{2} \cdot 2}{6 \cdot 2} - 2 \sqrt{2} + 7 \sqrt{2} - \frac{\sqrt{2}}{12}$

Step 2.3

Write $- 2 \sqrt{2}$ as a fraction with denominator $1$ .

$\frac{\sqrt{2} \cdot 2}{6 \cdot 2} + \frac{- 2 \sqrt{2}}{1} + 7 \sqrt{2} - \frac{\sqrt{2}}{12}$

Step 2.4

Multiply $\frac{- 2 \sqrt{2}}{1}$ by $\frac{12}{12}$ .

$\frac{\sqrt{2} \cdot 2}{6 \cdot 2} + \frac{- 2 \sqrt{2}}{1} \cdot \frac{12}{12} + 7 \sqrt{2} - \frac{\sqrt{2}}{12}$

Step 2.5

Multiply $\frac{- 2 \sqrt{2}}{1}$ by $\frac{12}{12}$ .

$\frac{\sqrt{2} \cdot 2}{6 \cdot 2} + \frac{- 2 \sqrt{2} \cdot 12}{12} + 7 \sqrt{2} - \frac{\sqrt{2}}{12}$

Step 2.6

Write $7 \sqrt{2}$ as a fraction with denominator $1$ .

$\frac{\sqrt{2} \cdot 2}{6 \cdot 2} + \frac{- 2 \sqrt{2} \cdot 12}{12} + \frac{7 \sqrt{2}}{1} - \frac{\sqrt{2}}{12}$

Step 2.7

Multiply $\frac{7 \sqrt{2}}{1}$ by $\frac{12}{12}$ .

$\frac{\sqrt{2} \cdot 2}{6 \cdot 2} + \frac{- 2 \sqrt{2} \cdot 12}{12} + \frac{7 \sqrt{2}}{1} \cdot \frac{12}{12} - \frac{\sqrt{2}}{12}$

Step 2.8

Multiply $\frac{7 \sqrt{2}}{1}$ by $\frac{12}{12}$ .

$\frac{\sqrt{2} \cdot 2}{6 \cdot 2} + \frac{- 2 \sqrt{2} \cdot 12}{12} + \frac{7 \sqrt{2} \cdot 12}{12} - \frac{\sqrt{2}}{12}$

Step 2.9

Reorder the factors of $6 \cdot 2$ .

$\frac{\sqrt{2} \cdot 2}{2 \cdot 6} + \frac{- 2 \sqrt{2} \cdot 12}{12} + \frac{7 \sqrt{2} \cdot 12}{12} - \frac{\sqrt{2}}{12}$

Step 2.10

Multiply $2$ by $6$ .

$\frac{\sqrt{2} \cdot 2}{12} + \frac{- 2 \sqrt{2} \cdot 12}{12} + \frac{7 \sqrt{2} \cdot 12}{12} - \frac{\sqrt{2}}{12}$

Step 3

Combine the numerators over the common denominator.

$\frac{\sqrt{2} \cdot 2 - 2 \sqrt{2} \cdot 12 + 7 \sqrt{2} \cdot 12 - \sqrt{2}}{12}$

Step 4

Simplify each term.

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Step 4.1

Move $2$ to the left of $\sqrt{2}$ .

$\frac{2 \cdot \sqrt{2} - 2 \sqrt{2} \cdot 12 + 7 \sqrt{2} \cdot 12 - \sqrt{2}}{12}$

Step 4.2

Multiply $12$ by $- 2$ .

$\frac{2 \sqrt{2} - 24 \sqrt{2} + 7 \sqrt{2} \cdot 12 - \sqrt{2}}{12}$

Step 4.3

Multiply $12$ by $7$ .

$\frac{2 \sqrt{2} - 24 \sqrt{2} + 84 \sqrt{2} - \sqrt{2}}{12}$

Step 5

Simplify by adding terms.

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Step 5.1

Subtract $24 \sqrt{2}$ from $2 \sqrt{2}$ .

$\frac{- 22 \sqrt{2} + 84 \sqrt{2} - \sqrt{2}}{12}$

Step 5.2

Add $- 22 \sqrt{2}$ and $84 \sqrt{2}$ .

$\frac{62 \sqrt{2} - \sqrt{2}}{12}$

Step 5.3

Subtract $\sqrt{2}$ from $62 \sqrt{2}$ .

$\frac{61 \sqrt{2}}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{61 \sqrt{2}}{12}$

Decimal Form:

$7.18891894 \dots$