Basic Math Examples

$- \frac{1}{2} \cdot \sqrt{245} + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $245$ as $7^{2} \cdot 5$ .

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Step 1.1.1

Factor $49$ out of $245$ .

$- \frac{1}{2} \cdot \sqrt{49 (5)} + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.1.2

Rewrite $49$ as $7^{2}$ .

$- \frac{1}{2} \cdot \sqrt{7^{2} \cdot 5} + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.2

Pull terms out from under the radical.

$- \frac{1}{2} \cdot (7 \sqrt{5}) + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.3

Multiply $- \frac{1}{2} (7 \sqrt{5})$ .

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Step 1.3.1

Multiply $7$ by $- 1$ .

$- 7 (\frac{1}{2}) \sqrt{5} + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.3.2

Combine $- 7$ and $\frac{1}{2}$ .

$\frac{- 7}{2} \sqrt{5} + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.3.3

Combine $\frac{- 7}{2}$ and $\sqrt{5}$ .

$\frac{- 7 \sqrt{5}}{2} + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.4

Move the negative in front of the fraction.

$- \frac{7 \sqrt{5}}{2} + \frac{7}{4} \cdot \sqrt{125} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.5

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 1.5.1

Factor $25$ out of $125$ .

$- \frac{7 \sqrt{5}}{2} + \frac{7}{4} \cdot \sqrt{25 (5)} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.5.2

Rewrite $25$ as $5^{2}$ .

$- \frac{7 \sqrt{5}}{2} + \frac{7}{4} \cdot \sqrt{5^{2} \cdot 5} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.6

Pull terms out from under the radical.

$- \frac{7 \sqrt{5}}{2} + \frac{7}{4} \cdot (5 \sqrt{5}) + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.7

Multiply $\frac{7}{4} (5 \sqrt{5})$ .

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Step 1.7.1

Combine $5$ and $\frac{7}{4}$ .

$- \frac{7 \sqrt{5}}{2} + \frac{5 \cdot 7}{4} \sqrt{5} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.7.2

Multiply $5$ by $7$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35}{4} \sqrt{5} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.7.3

Combine $\frac{35}{4}$ and $\sqrt{5}$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{7}{5} \cdot \sqrt{180} + \sqrt{245}$

Step 1.8

Rewrite $180$ as $6^{2} \cdot 5$ .

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Step 1.8.1

Factor $36$ out of $180$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{7}{5} \cdot \sqrt{36 (5)} + \sqrt{245}$

Step 1.8.2

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

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{7}{5} \cdot \sqrt{6^{2} \cdot 5} + \sqrt{245}$

Step 1.9

Pull terms out from under the radical.

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{7}{5} \cdot (6 \sqrt{5}) + \sqrt{245}$

Step 1.10

Multiply $\frac{7}{5} (6 \sqrt{5})$ .

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Step 1.10.1

Combine $6$ and $\frac{7}{5}$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{6 \cdot 7}{5} \sqrt{5} + \sqrt{245}$

Step 1.10.2

Multiply $6$ by $7$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{42}{5} \sqrt{5} + \sqrt{245}$

Step 1.10.3

Combine $\frac{42}{5}$ and $\sqrt{5}$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{42 \sqrt{5}}{5} + \sqrt{245}$

Step 1.11

Rewrite $245$ as $7^{2} \cdot 5$ .

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Step 1.11.1

Factor $49$ out of $245$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{42 \sqrt{5}}{5} + \sqrt{49 (5)}$

Step 1.11.2

Rewrite $49$ as $7^{2}$ .

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{42 \sqrt{5}}{5} + \sqrt{7^{2} \cdot 5}$

Step 1.12

Pull terms out from under the radical.

$- \frac{7 \sqrt{5}}{2} + \frac{35 \sqrt{5}}{4} + \frac{42 \sqrt{5}}{5} + 7 \sqrt{5}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{7 \sqrt{5}}{2}$ by $\frac{10}{10}$ .

$- (\frac{7 \sqrt{5}}{2} \cdot \frac{10}{10}) + \frac{35 \sqrt{5}}{4} + \frac{42 \sqrt{5}}{5} + 7 \sqrt{5}$

Step 2.2

Multiply $\frac{7 \sqrt{5}}{2}$ by $\frac{10}{10}$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5}}{4} + \frac{42 \sqrt{5}}{5} + 7 \sqrt{5}$

Step 2.3

Multiply $\frac{35 \sqrt{5}}{4}$ by $\frac{5}{5}$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5}}{4} \cdot \frac{5}{5} + \frac{42 \sqrt{5}}{5} + 7 \sqrt{5}$

Step 2.4

Multiply $\frac{35 \sqrt{5}}{4}$ by $\frac{5}{5}$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5} \cdot 5}{4 \cdot 5} + \frac{42 \sqrt{5}}{5} + 7 \sqrt{5}$

Step 2.5

Multiply $\frac{42 \sqrt{5}}{5}$ by $\frac{4}{4}$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5} \cdot 5}{4 \cdot 5} + \frac{42 \sqrt{5}}{5} \cdot \frac{4}{4} + 7 \sqrt{5}$

Step 2.6

Multiply $\frac{42 \sqrt{5}}{5}$ by $\frac{4}{4}$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5} \cdot 5}{4 \cdot 5} + \frac{42 \sqrt{5} \cdot 4}{5 \cdot 4} + 7 \sqrt{5}$

Step 2.7

Write $7 \sqrt{5}$ as a fraction with denominator $1$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5} \cdot 5}{4 \cdot 5} + \frac{42 \sqrt{5} \cdot 4}{5 \cdot 4} + \frac{7 \sqrt{5}}{1}$

Step 2.8

Multiply $\frac{7 \sqrt{5}}{1}$ by $\frac{20}{20}$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5} \cdot 5}{4 \cdot 5} + \frac{42 \sqrt{5} \cdot 4}{5 \cdot 4} + \frac{7 \sqrt{5}}{1} \cdot \frac{20}{20}$

Step 2.9

Multiply $\frac{7 \sqrt{5}}{1}$ by $\frac{20}{20}$ .

$- \frac{7 \sqrt{5} \cdot 10}{2 \cdot 10} + \frac{35 \sqrt{5} \cdot 5}{4 \cdot 5} + \frac{42 \sqrt{5} \cdot 4}{5 \cdot 4} + \frac{7 \sqrt{5} \cdot 20}{20}$

Step 2.10

Multiply $2$ by $10$ .

$- \frac{7 \sqrt{5} \cdot 10}{20} + \frac{35 \sqrt{5} \cdot 5}{4 \cdot 5} + \frac{42 \sqrt{5} \cdot 4}{5 \cdot 4} + \frac{7 \sqrt{5} \cdot 20}{20}$

Step 2.11

Multiply $4$ by $5$ .

$- \frac{7 \sqrt{5} \cdot 10}{20} + \frac{35 \sqrt{5} \cdot 5}{20} + \frac{42 \sqrt{5} \cdot 4}{5 \cdot 4} + \frac{7 \sqrt{5} \cdot 20}{20}$

Step 2.12

Reorder the factors of $5 \cdot 4$ .

$- \frac{7 \sqrt{5} \cdot 10}{20} + \frac{35 \sqrt{5} \cdot 5}{20} + \frac{42 \sqrt{5} \cdot 4}{4 \cdot 5} + \frac{7 \sqrt{5} \cdot 20}{20}$

Step 2.13

Multiply $4$ by $5$ .

$- \frac{7 \sqrt{5} \cdot 10}{20} + \frac{35 \sqrt{5} \cdot 5}{20} + \frac{42 \sqrt{5} \cdot 4}{20} + \frac{7 \sqrt{5} \cdot 20}{20}$

Step 3

Combine the numerators over the common denominator.

$\frac{- 7 \sqrt{5} \cdot 10 + 35 \sqrt{5} \cdot 5 + 42 \sqrt{5} \cdot 4 + 7 \sqrt{5} \cdot 20}{20}$

Step 4

Simplify each term.

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Step 4.1

Multiply $10$ by $- 7$ .

$\frac{- 70 \sqrt{5} + 35 \sqrt{5} \cdot 5 + 42 \sqrt{5} \cdot 4 + 7 \sqrt{5} \cdot 20}{20}$

Step 4.2

Multiply $5$ by $35$ .

$\frac{- 70 \sqrt{5} + 175 \sqrt{5} + 42 \sqrt{5} \cdot 4 + 7 \sqrt{5} \cdot 20}{20}$

Step 4.3

Multiply $4$ by $42$ .

$\frac{- 70 \sqrt{5} + 175 \sqrt{5} + 168 \sqrt{5} + 7 \sqrt{5} \cdot 20}{20}$

Step 4.4

Multiply $20$ by $7$ .

$\frac{- 70 \sqrt{5} + 175 \sqrt{5} + 168 \sqrt{5} + 140 \sqrt{5}}{20}$

Step 5

Simplify by adding terms.

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Step 5.1

Add $- 70 \sqrt{5}$ and $175 \sqrt{5}$ .

$\frac{105 \sqrt{5} + 168 \sqrt{5} + 140 \sqrt{5}}{20}$

Step 5.2

Add $105 \sqrt{5}$ and $168 \sqrt{5}$ .

$\frac{273 \sqrt{5} + 140 \sqrt{5}}{20}$

Step 5.3

Add $273 \sqrt{5}$ and $140 \sqrt{5}$ .

$\frac{413 \sqrt{5}}{20}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{413 \sqrt{5}}{20}$

Decimal Form:

$46.17480373 \dots$