Basic Math Examples

$\frac{1}{3} \cdot \sqrt{72} + \frac{1}{4} \cdot \sqrt{32} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $36$ out of $72$ .

$\frac{1}{3} \cdot \sqrt{36 (2)} + \frac{1}{4} \cdot \sqrt{32} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.1.2

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

$\frac{1}{3} \cdot \sqrt{6^{2} \cdot 2} + \frac{1}{4} \cdot \sqrt{32} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.2

Pull terms out from under the radical.

$\frac{1}{3} \cdot (6 \sqrt{2}) + \frac{1}{4} \cdot \sqrt{32} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.3

Cancel the common factor of $3$ .

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Step 1.3.1

Factor $3$ out of $6 \sqrt{2}$ .

$\frac{1}{3} \cdot (3 (2 \sqrt{2})) + \frac{1}{4} \cdot \sqrt{32} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.3.2

Cancel the common factor.

$\frac{1}{3} \cdot (3 (2 \sqrt{2})) + \frac{1}{4} \cdot \sqrt{32} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.3.3

Rewrite the expression.

$2 \sqrt{2} + \frac{1}{4} \cdot \sqrt{32} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.4.1

Factor $16$ out of $32$ .

$2 \sqrt{2} + \frac{1}{4} \cdot \sqrt{16 (2)} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.4.2

Rewrite $16$ as $4^{2}$ .

$2 \sqrt{2} + \frac{1}{4} \cdot \sqrt{4^{2} \cdot 2} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.5

Pull terms out from under the radical.

$2 \sqrt{2} + \frac{1}{4} \cdot (4 \sqrt{2}) - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.6

Cancel the common factor of $4$ .

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Step 1.6.1

Factor $4$ out of $4 \sqrt{2}$ .

$2 \sqrt{2} + \frac{1}{4} \cdot (4 (\sqrt{2})) - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.6.2

Cancel the common factor.

$2 \sqrt{2} + \frac{1}{4} \cdot (4 \sqrt{2}) - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.6.3

Rewrite the expression.

$2 \sqrt{2} + \sqrt{2} - \frac{1}{2} \cdot \sqrt{72} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.7

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

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Step 1.7.1

Factor $36$ out of $72$ .

$2 \sqrt{2} + \sqrt{2} - \frac{1}{2} \cdot \sqrt{36 (2)} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.7.2

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

$2 \sqrt{2} + \sqrt{2} - \frac{1}{2} \cdot \sqrt{6^{2} \cdot 2} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.8

Pull terms out from under the radical.

$2 \sqrt{2} + \sqrt{2} - \frac{1}{2} \cdot (6 \sqrt{2}) + \frac{7}{12} \cdot \sqrt{18}$

Step 1.9

Cancel the common factor of $2$ .

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Step 1.9.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2 \sqrt{2} + \sqrt{2} + \frac{- 1}{2} \cdot (6 \sqrt{2}) + \frac{7}{12} \cdot \sqrt{18}$

Step 1.9.2

Factor $2$ out of $6 \sqrt{2}$ .

$2 \sqrt{2} + \sqrt{2} + \frac{- 1}{2} \cdot (2 (3 \sqrt{2})) + \frac{7}{12} \cdot \sqrt{18}$

Step 1.9.3

Cancel the common factor.

$2 \sqrt{2} + \sqrt{2} + \frac{- 1}{2} \cdot (2 (3 \sqrt{2})) + \frac{7}{12} \cdot \sqrt{18}$

Step 1.9.4

Rewrite the expression.

$2 \sqrt{2} + \sqrt{2} - 1 \cdot (3 \sqrt{2}) + \frac{7}{12} \cdot \sqrt{18}$

Step 1.10

Multiply $3$ by $- 1$ .

$2 \sqrt{2} + \sqrt{2} - 3 \cdot \sqrt{2} + \frac{7}{12} \cdot \sqrt{18}$

Step 1.11

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

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Step 1.11.1

Factor $9$ out of $18$ .

$2 \sqrt{2} + \sqrt{2} - 3 \sqrt{2} + \frac{7}{12} \cdot \sqrt{9 (2)}$

Step 1.11.2

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

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

Step 1.12

Pull terms out from under the radical.

$2 \sqrt{2} + \sqrt{2} - 3 \sqrt{2} + \frac{7}{12} \cdot (3 \sqrt{2})$

Step 1.13

Cancel the common factor of $3$ .

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Step 1.13.1

Factor $3$ out of $12$ .

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

Step 1.13.2

Factor $3$ out of $3 \sqrt{2}$ .

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

Step 1.13.3

Cancel the common factor.

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

Step 1.13.4

Rewrite the expression.

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

Step 1.14

Combine $\frac{7}{4}$ and $\sqrt{2}$ .

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

Step 2

Find the common denominator.

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Step 2.1

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

$\frac{2 \sqrt{2}}{1} + \sqrt{2} - 3 \sqrt{2} + \frac{7 \sqrt{2}}{4}$

Step 2.2

Multiply $\frac{2 \sqrt{2}}{1}$ by $\frac{4}{4}$ .

$\frac{2 \sqrt{2}}{1} \cdot \frac{4}{4} + \sqrt{2} - 3 \sqrt{2} + \frac{7 \sqrt{2}}{4}$

Step 2.3

Multiply $\frac{2 \sqrt{2}}{1}$ by $\frac{4}{4}$ .

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

Step 2.4

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

$\frac{2 \sqrt{2} \cdot 4}{4} + \frac{\sqrt{2}}{1} - 3 \sqrt{2} + \frac{7 \sqrt{2}}{4}$

Step 2.5

Multiply $\frac{\sqrt{2}}{1}$ by $\frac{4}{4}$ .

$\frac{2 \sqrt{2} \cdot 4}{4} + \frac{\sqrt{2}}{1} \cdot \frac{4}{4} - 3 \sqrt{2} + \frac{7 \sqrt{2}}{4}$

Step 2.6

Multiply $\frac{\sqrt{2}}{1}$ by $\frac{4}{4}$ .

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

Step 2.7

Write $- 3 \sqrt{2}$ as a fraction with denominator $1$ .

$\frac{2 \sqrt{2} \cdot 4}{4} + \frac{\sqrt{2} \cdot 4}{4} + \frac{- 3 \sqrt{2}}{1} + \frac{7 \sqrt{2}}{4}$

Step 2.8

Multiply $\frac{- 3 \sqrt{2}}{1}$ by $\frac{4}{4}$ .

$\frac{2 \sqrt{2} \cdot 4}{4} + \frac{\sqrt{2} \cdot 4}{4} + \frac{- 3 \sqrt{2}}{1} \cdot \frac{4}{4} + \frac{7 \sqrt{2}}{4}$

Step 2.9

Multiply $\frac{- 3 \sqrt{2}}{1}$ by $\frac{4}{4}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $4$ by $2$ .

$\frac{8 \sqrt{2} + \sqrt{2} \cdot 4 - 3 \sqrt{2} \cdot 4 + 7 \sqrt{2}}{4}$

Step 4.2

Move $4$ to the left of $\sqrt{2}$ .

$\frac{8 \sqrt{2} + 4 \cdot \sqrt{2} - 3 \sqrt{2} \cdot 4 + 7 \sqrt{2}}{4}$

Step 4.3

Multiply $4$ by $- 3$ .

$\frac{8 \sqrt{2} + 4 \sqrt{2} - 12 \sqrt{2} + 7 \sqrt{2}}{4}$

Step 5

Simplify by adding terms.

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Step 5.1

Add $8 \sqrt{2}$ and $4 \sqrt{2}$ .

$\frac{12 \sqrt{2} - 12 \sqrt{2} + 7 \sqrt{2}}{4}$

Step 5.2

Subtract $12 \sqrt{2}$ from $12 \sqrt{2}$ .

$\frac{0 + 7 \sqrt{2}}{4}$

Step 5.3

Add $0$ and $7 \sqrt{2}$ .

$\frac{7 \sqrt{2}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{7 \sqrt{2}}{4}$

Decimal Form:

$2.47487373 \dots$