Algebra Examples

\sqrt{121} + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}

$\sqrt{121} + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}$

Step 1

Simplify each term.

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Step 1.1

Rewrite

121

$121$ as

11^{2}

$11^{2}$ .

\sqrt{11^{2}} + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}

$\sqrt{11^{2}} + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}$

Step 1.2

Pull terms out from under the radical.

| 11 | + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}

$| 11 | + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}$

Step 1.3

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

11

$11$ is

11

$11$ .

11 + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}

$11 + \sqrt{- 28} - \sqrt{49} - \sqrt{- 112}$

Step 1.4

Rewrite

- 28

$- 28$ as

- 1 (28)

$- 1 (28)$ .

11 + \sqrt{- 1 (28)} - \sqrt{49} - \sqrt{- 112}

$11 + \sqrt{- 1 (28)} - \sqrt{49} - \sqrt{- 112}$

Step 1.5

Rewrite

\sqrt{- 1 (28)}

$\sqrt{- 1 (28)}$ as

\sqrt{- 1} \cdot \sqrt{28}

$\sqrt{- 1} \cdot \sqrt{28}$ .

11 + \sqrt{- 1} \cdot \sqrt{28} - \sqrt{49} - \sqrt{- 112}

$11 + \sqrt{- 1} \cdot \sqrt{28} - \sqrt{49} - \sqrt{- 112}$

Step 1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

11 + i \cdot \sqrt{28} - \sqrt{49} - \sqrt{- 112}

$11 + i \cdot \sqrt{28} - \sqrt{49} - \sqrt{- 112}$

Step 1.7

Rewrite

28

$28$ as

2^{2} \cdot 7

$2^{2} \cdot 7$ .

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Step 1.7.1

Factor

4

$4$ out of

28

$28$ .

11 + i \cdot \sqrt{4 (7)} - \sqrt{49} - \sqrt{- 112}

$11 + i \cdot \sqrt{4 (7)} - \sqrt{49} - \sqrt{- 112}$

Step 1.7.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

11 + i \cdot \sqrt{2^{2} \cdot 7} - \sqrt{49} - \sqrt{- 112}

$11 + i \cdot \sqrt{2^{2} \cdot 7} - \sqrt{49} - \sqrt{- 112}$

11 + i \cdot \sqrt{2^{2} \cdot 7} - \sqrt{49} - \sqrt{- 112}

$11 + i \cdot \sqrt{2^{2} \cdot 7} - \sqrt{49} - \sqrt{- 112}$

Step 1.8

Pull terms out from under the radical.

11 + i \cdot (| 2 | \sqrt{7}) - \sqrt{49} - \sqrt{- 112}

$11 + i \cdot (| 2 | \sqrt{7}) - \sqrt{49} - \sqrt{- 112}$

Step 1.9

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

2

$2$ is

2

$2$ .

11 + i \cdot (2 \sqrt{7}) - \sqrt{49} - \sqrt{- 112}

$11 + i \cdot (2 \sqrt{7}) - \sqrt{49} - \sqrt{- 112}$

Step 1.10

Move

2

$2$ to the left of

i

$i$ .

11 + 2 \cdot i \sqrt{7} - \sqrt{49} - \sqrt{- 112}

$11 + 2 \cdot i \sqrt{7} - \sqrt{49} - \sqrt{- 112}$

Step 1.11

Rewrite

49

$49$ as

7^{2}

$7^{2}$ .

11 + 2 i \sqrt{7} - \sqrt{7^{2}} - \sqrt{- 112}

$11 + 2 i \sqrt{7} - \sqrt{7^{2}} - \sqrt{- 112}$

Step 1.12

Pull terms out from under the radical.

11 + 2 i \sqrt{7} - | 7 | - \sqrt{- 112}

$11 + 2 i \sqrt{7} - | 7 | - \sqrt{- 112}$

Step 1.13

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

7

$7$ is

7

$7$ .

11 + 2 i \sqrt{7} - 1 \cdot 7 - \sqrt{- 112}

$11 + 2 i \sqrt{7} - 1 \cdot 7 - \sqrt{- 112}$

Step 1.14

Multiply

- 1

$- 1$ by

7

$7$ .

11 + 2 i \sqrt{7} - 7 - \sqrt{- 112}

$11 + 2 i \sqrt{7} - 7 - \sqrt{- 112}$

Step 1.15

Rewrite

- 112

$- 112$ as

- 1 (112)

$- 1 (112)$ .

11 + 2 i \sqrt{7} - 7 - \sqrt{- 1 (112)}

$11 + 2 i \sqrt{7} - 7 - \sqrt{- 1 (112)}$

Step 1.16

Rewrite

\sqrt{- 1 (112)}

$\sqrt{- 1 (112)}$ as

\sqrt{- 1} \cdot \sqrt{112}

$\sqrt{- 1} \cdot \sqrt{112}$ .

11 + 2 i \sqrt{7} - 7 - (\sqrt{- 1} \cdot \sqrt{112})

$11 + 2 i \sqrt{7} - 7 - (\sqrt{- 1} \cdot \sqrt{112})$

Step 1.17

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{112})

$11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{112})$

Step 1.18

Rewrite

112

$112$ as

4^{2} \cdot 7

$4^{2} \cdot 7$ .

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Step 1.18.1

Factor

16

$16$ out of

112

$112$ .

11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{16 (7)})

$11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{16 (7)})$

Step 1.18.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{4^{2} \cdot 7})

$11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{4^{2} \cdot 7})$

11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{4^{2} \cdot 7})

$11 + 2 i \sqrt{7} - 7 - (i \cdot \sqrt{4^{2} \cdot 7})$

Step 1.19

Pull terms out from under the radical.

11 + 2 i \sqrt{7} - 7 - (i \cdot (| 4 | \sqrt{7}))

$11 + 2 i \sqrt{7} - 7 - (i \cdot (| 4 | \sqrt{7}))$

Step 1.20

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

4

$4$ is

4

$4$ .

11 + 2 i \sqrt{7} - 7 - (i \cdot (4 \sqrt{7}))

$11 + 2 i \sqrt{7} - 7 - (i \cdot (4 \sqrt{7}))$

Step 1.21

Move

4

$4$ to the left of

i

$i$ .

11 + 2 i \sqrt{7} - 7 - (4 \cdot i \sqrt{7})

$11 + 2 i \sqrt{7} - 7 - (4 \cdot i \sqrt{7})$

Step 1.22

Multiply

4

$4$ by

- 1

$- 1$ .

11 + 2 i \sqrt{7} - 7 - 4 i \sqrt{7}

$11 + 2 i \sqrt{7} - 7 - 4 i \sqrt{7}$

11 + 2 i \sqrt{7} - 7 - 4 i \sqrt{7}

$11 + 2 i \sqrt{7} - 7 - 4 i \sqrt{7}$

Step 2

Subtract

7

$7$ from

11

$11$ .

2 i \sqrt{7} + 4 - 4 i \sqrt{7}

$2 i \sqrt{7} + 4 - 4 i \sqrt{7}$

Step 3

Subtract

4 i \sqrt{7}

$4 i \sqrt{7}$ from

2 i \sqrt{7}

$2 i \sqrt{7}$ .

- 2 i \sqrt{7} + 4

$- 2 i \sqrt{7} + 4$

Step 4

Reorder

- 2 i \sqrt{7}

$- 2 i \sqrt{7}$ and

4

$4$ .

4 - 2 i \sqrt{7}

$4 - 2 i \sqrt{7}$