Algebra Examples

$\sqrt{49} - \sqrt{- 147} + \sqrt{1} + \sqrt{- 108}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{7^{2}} - \sqrt{- 147} + \sqrt{1} + \sqrt{- 108}$

Step 1.2

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

$7 - \sqrt{- 147} + \sqrt{1} + \sqrt{- 108}$

Step 1.3

Rewrite $- 147$ as $- 1 (147)$ .

$7 - \sqrt{- 1 (147)} + \sqrt{1} + \sqrt{- 108}$

Step 1.4

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$7 - (\sqrt{- 1} \cdot \sqrt{147}) + \sqrt{1} + \sqrt{- 108}$

Step 1.5

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

$7 - (i \cdot \sqrt{147}) + \sqrt{1} + \sqrt{- 108}$

Step 1.6

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 1.6.1

Factor $49$ out of $147$ .

$7 - (i \cdot \sqrt{49 (3)}) + \sqrt{1} + \sqrt{- 108}$

Step 1.6.2

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

$7 - (i \cdot \sqrt{7^{2} \cdot 3}) + \sqrt{1} + \sqrt{- 108}$

Step 1.7

Pull terms out from under the radical.

$7 - (i \cdot (7 \sqrt{3})) + \sqrt{1} + \sqrt{- 108}$

Step 1.8

Move $7$ to the left of $i$ .

$7 - (7 \cdot i \sqrt{3}) + \sqrt{1} + \sqrt{- 108}$

Step 1.9

Multiply $7$ by $- 1$ .

$7 - 7 (i \sqrt{3}) + \sqrt{1} + \sqrt{- 108}$

Step 1.10

Any root of $1$ is $1$ .

$7 - 7 i \sqrt{3} + 1 + \sqrt{- 108}$

Step 1.11

Rewrite $- 108$ as $- 1 (108)$ .

$7 - 7 i \sqrt{3} + 1 + \sqrt{- 1 (108)}$

Step 1.12

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$7 - 7 i \sqrt{3} + 1 + \sqrt{- 1} \cdot \sqrt{108}$

Step 1.13

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

$7 - 7 i \sqrt{3} + 1 + i \cdot \sqrt{108}$

Step 1.14

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 1.14.1

Factor $36$ out of $108$ .

$7 - 7 i \sqrt{3} + 1 + i \cdot \sqrt{36 (3)}$

Step 1.14.2

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

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

Step 1.15

Pull terms out from under the radical.

$7 - 7 i \sqrt{3} + 1 + i \cdot (6 \sqrt{3})$

Step 1.16

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

$7 - 7 i \sqrt{3} + 1 + 6 i \sqrt{3}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $7$ and $1$ .

$- 7 i \sqrt{3} + 8 + 6 i \sqrt{3}$

Step 2.2

Add $- 7 i \sqrt{3}$ and $6 i \sqrt{3}$ .

$- i \sqrt{3} + 8$

Step 2.3

Reorder $- i \sqrt{3}$ and $8$ .

$8 - i \sqrt{3}$