Basic Math Examples

\sqrt{112} - \sqrt{48} + (4 \sqrt{7} + \sqrt{- 147})

$\sqrt{112} - \sqrt{48} + (4 \sqrt{7} + \sqrt{- 147})$

Step 1

Remove parentheses.

\sqrt{112} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}

$\sqrt{112} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}$

Step 2

Simplify each term.

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Step 2.1

Rewrite

112

$112$ as

4^{2} \cdot 7

$4^{2} \cdot 7$ .

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Step 2.1.1

Factor

16

$16$ out of

112

$112$ .

\sqrt{16 (7)} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}

$\sqrt{16 (7)} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}$

Step 2.1.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

\sqrt{4^{2} \cdot 7} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}

$\sqrt{4^{2} \cdot 7} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}$

\sqrt{4^{2} \cdot 7} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}

$\sqrt{4^{2} \cdot 7} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}$

Step 2.2

Pull terms out from under the radical.

4 \sqrt{7} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}

$4 \sqrt{7} - \sqrt{48} + 4 \sqrt{7} + \sqrt{- 147}$

Step 2.3

Rewrite

48

$48$ as

4^{2} \cdot 3

$4^{2} \cdot 3$ .

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Step 2.3.1

Factor

16

$16$ out of

48

$48$ .

4 \sqrt{7} - \sqrt{16 (3)} + 4 \sqrt{7} + \sqrt{- 147}

$4 \sqrt{7} - \sqrt{16 (3)} + 4 \sqrt{7} + \sqrt{- 147}$

Step 2.3.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

4 \sqrt{7} - \sqrt{4^{2} \cdot 3} + 4 \sqrt{7} + \sqrt{- 147}

$4 \sqrt{7} - \sqrt{4^{2} \cdot 3} + 4 \sqrt{7} + \sqrt{- 147}$

4 \sqrt{7} - \sqrt{4^{2} \cdot 3} + 4 \sqrt{7} + \sqrt{- 147}

$4 \sqrt{7} - \sqrt{4^{2} \cdot 3} + 4 \sqrt{7} + \sqrt{- 147}$

Step 2.4

Pull terms out from under the radical.

4 \sqrt{7} - (4 \sqrt{3}) + 4 \sqrt{7} + \sqrt{- 147}

$4 \sqrt{7} - (4 \sqrt{3}) + 4 \sqrt{7} + \sqrt{- 147}$

Step 2.5

Multiply

4

$4$ by

- 1

$- 1$ .

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + \sqrt{- 147}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + \sqrt{- 147}$

Step 2.6

Rewrite

- 147

$- 147$ as

- 1 (147)

$- 1 (147)$ .

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + \sqrt{- 1 (147)}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + \sqrt{- 1 (147)}$

Step 2.7

Rewrite

\sqrt{- 1 (147)}

$\sqrt{- 1 (147)}$ as

\sqrt{- 1} \cdot \sqrt{147}

$\sqrt{- 1} \cdot \sqrt{147}$ .

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + \sqrt{- 1} \cdot \sqrt{147}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + \sqrt{- 1} \cdot \sqrt{147}$

Step 2.8

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{147}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{147}$

Step 2.9

Rewrite

147

$147$ as

7^{2} \cdot 3

$7^{2} \cdot 3$ .

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Step 2.9.1

Factor

49

$49$ out of

147

$147$ .

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{49 (3)}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{49 (3)}$

Step 2.9.2

Rewrite

49

$49$ as

7^{2}

$7^{2}$ .

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{7^{2} \cdot 3}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{7^{2} \cdot 3}$

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{7^{2} \cdot 3}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot \sqrt{7^{2} \cdot 3}$

Step 2.10

Pull terms out from under the radical.

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot (7 \sqrt{3})

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + i \cdot (7 \sqrt{3})$

Step 2.11

Move

7

$7$ to the left of

i

$i$ .

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + 7 i \sqrt{3}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + 7 i \sqrt{3}$

4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + 7 i \sqrt{3}

$4 \sqrt{7} - 4 \sqrt{3} + 4 \sqrt{7} + 7 i \sqrt{3}$

Step 3

Add

4 \sqrt{7}

$4 \sqrt{7}$ and

4 \sqrt{7}

$4 \sqrt{7}$ .

8 \sqrt{7} - 4 \sqrt{3} + 7 i \sqrt{3}

$8 \sqrt{7} - 4 \sqrt{3} + 7 i \sqrt{3}$