Algebra Examples

$- \sqrt{25} + \sqrt{- 40} + \sqrt{81} - \sqrt{- 10}$

Step 1

Simplify each term.

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Step 1.1

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

$- \sqrt{5^{2}} + \sqrt{- 40} + \sqrt{81} - \sqrt{- 10}$

Step 1.2

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

$- 1 \cdot 5 + \sqrt{- 40} + \sqrt{81} - \sqrt{- 10}$

Step 1.3

Multiply $- 1$ by $5$ .

$- 5 + \sqrt{- 40} + \sqrt{81} - \sqrt{- 10}$

Step 1.4

Rewrite $- 40$ as $- 1 (40)$ .

$- 5 + \sqrt{- 1 (40)} + \sqrt{81} - \sqrt{- 10}$

Step 1.5

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$- 5 + \sqrt{- 1} \cdot \sqrt{40} + \sqrt{81} - \sqrt{- 10}$

Step 1.6

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

$- 5 + i \cdot \sqrt{40} + \sqrt{81} - \sqrt{- 10}$

Step 1.7

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 1.7.1

Factor $4$ out of $40$ .

$- 5 + i \cdot \sqrt{4 (10)} + \sqrt{81} - \sqrt{- 10}$

Step 1.7.2

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

$- 5 + i \cdot \sqrt{2^{2} \cdot 10} + \sqrt{81} - \sqrt{- 10}$

Step 1.8

Pull terms out from under the radical.

$- 5 + i \cdot (2 \sqrt{10}) + \sqrt{81} - \sqrt{- 10}$

Step 1.9

Move $2$ to the left of $i$ .

$- 5 + 2 \cdot i \sqrt{10} + \sqrt{81} - \sqrt{- 10}$

Step 1.10

Rewrite $81$ as $9^{2}$ .

$- 5 + 2 i \sqrt{10} + \sqrt{9^{2}} - \sqrt{- 10}$

Step 1.11

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

$- 5 + 2 i \sqrt{10} + 9 - \sqrt{- 10}$

Step 1.12

Rewrite $- 10$ as $- 1 (10)$ .

$- 5 + 2 i \sqrt{10} + 9 - \sqrt{- 1 (10)}$

Step 1.13

Rewrite $\sqrt{- 1 (10)}$ as $\sqrt{- 1} \cdot \sqrt{10}$ .

$- 5 + 2 i \sqrt{10} + 9 - (\sqrt{- 1} \cdot \sqrt{10})$

Step 1.14

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

$- 5 + 2 i \sqrt{10} + 9 - i \sqrt{10}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 5$ and $9$ .

$2 i \sqrt{10} + 4 - i \sqrt{10}$

Step 2.2

Subtract $i \sqrt{10}$ from $2 i \sqrt{10}$ .

$1 i \sqrt{10} + 4$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $i$ by $1$ .

$i \sqrt{10} + 4$

Step 2.3.2

Reorder $i \sqrt{10}$ and $4$ .

$4 + i \sqrt{10}$