Trigonometry Examples

$\sqrt{100} - \sqrt{- 175} + \sqrt{25} - \sqrt{- 252}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2}} - \sqrt{- 175} + \sqrt{25} - \sqrt{- 252}$

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

$10 - \sqrt{- 175} + \sqrt{25} - \sqrt{- 252}$

Step 1.3

Rewrite $- 175$ as $- 1 (175)$ .

$10 - \sqrt{- 1 (175)} + \sqrt{25} - \sqrt{- 252}$

Step 1.4

Rewrite $\sqrt{- 1 (175)}$ as $\sqrt{- 1} \cdot \sqrt{175}$ .

$10 - (\sqrt{- 1} \cdot \sqrt{175}) + \sqrt{25} - \sqrt{- 252}$

Step 1.5

Rewrite $\sqrt{- 1}$ as $i$ .

$10 - (i \cdot \sqrt{175}) + \sqrt{25} - \sqrt{- 252}$

Step 1.6

Rewrite $175$ as $5^{2} \cdot 7$ .

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Step 1.6.1

Factor $25$ out of $175$ .

$10 - (i \cdot \sqrt{25 (7)}) + \sqrt{25} - \sqrt{- 252}$

Step 1.6.2

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

$10 - (i \cdot \sqrt{5^{2} \cdot 7}) + \sqrt{25} - \sqrt{- 252}$

Step 1.7

Pull terms out from under the radical.

$10 - (i \cdot (5 \sqrt{7})) + \sqrt{25} - \sqrt{- 252}$

Step 1.8

Move $5$ to the left of $i$ .

$10 - (5 \cdot i \sqrt{7}) + \sqrt{25} - \sqrt{- 252}$

Step 1.9

Multiply $5$ by $- 1$ .

$10 - 5 (i \sqrt{7}) + \sqrt{25} - \sqrt{- 252}$

Step 1.10

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

$10 - 5 i \sqrt{7} + \sqrt{5^{2}} - \sqrt{- 252}$

Step 1.11

Pull terms out from under the radical, assuming positive real numbers.

$10 - 5 i \sqrt{7} + 5 - \sqrt{- 252}$

Step 1.12

Rewrite $- 252$ as $- 1 (252)$ .

$10 - 5 i \sqrt{7} + 5 - \sqrt{- 1 (252)}$

Step 1.13

Rewrite $\sqrt{- 1 (252)}$ as $\sqrt{- 1} \cdot \sqrt{252}$ .

$10 - 5 i \sqrt{7} + 5 - (\sqrt{- 1} \cdot \sqrt{252})$

Step 1.14

Rewrite $\sqrt{- 1}$ as $i$ .

$10 - 5 i \sqrt{7} + 5 - (i \cdot \sqrt{252})$

Step 1.15

Rewrite $252$ as $6^{2} \cdot 7$ .

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Step 1.15.1

Factor $36$ out of $252$ .

$10 - 5 i \sqrt{7} + 5 - (i \cdot \sqrt{36 (7)})$

Step 1.15.2

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

$10 - 5 i \sqrt{7} + 5 - (i \cdot \sqrt{6^{2} \cdot 7})$

Step 1.16

Pull terms out from under the radical.

$10 - 5 i \sqrt{7} + 5 - (i \cdot (6 \sqrt{7}))$

Step 1.17

Move $6$ to the left of $i$ .

$10 - 5 i \sqrt{7} + 5 - (6 \cdot i \sqrt{7})$

Step 1.18

Multiply $6$ by $- 1$ .

$10 - 5 i \sqrt{7} + 5 - 6 i \sqrt{7}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $10$ and $5$ .

$- 5 i \sqrt{7} + 15 - 6 i \sqrt{7}$

Step 2.2

Subtract $6 i \sqrt{7}$ from $- 5 i \sqrt{7}$ .

$- 11 i \sqrt{7} + 15$

Step 2.3

Reorder $- 11 i \sqrt{7}$ and $15$ .

$15 - 11 i \sqrt{7}$