Algebra Examples

$(4 + \sqrt{- 49}) - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324}) = a + b i$

Step 1

Rewrite the equation as $a + b i = 4 + \sqrt{- 49} - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$ .

$a + b i = 4 + \sqrt{- 49} - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$

Step 2

Simplify $4 + \sqrt{- 49} - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $- 49$ as $- 1 (49)$ .

$a + b i = 4 + \sqrt{- 1 (49)} - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$

Step 2.1.2

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$a + b i = 4 + \sqrt{- 1} \cdot \sqrt{49} - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$

Step 2.1.3

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

$a + b i = 4 + i \cdot \sqrt{49} - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$

Step 2.1.4

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

$a + b i = 4 + i \cdot \sqrt{7^{2}} - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$

Step 2.1.5

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

$a + b i = 4 + i \cdot 7 - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$

Step 2.1.6

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

$a + b i = 4 + 7 \cdot i - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 324})$

Step 2.1.7

Rewrite $- 324$ as $- 1 (324)$ .

$a + b i = 4 + 7 i - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 1 (324)})$

Step 2.1.8

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$a + b i = 4 + 7 i - 2 (\sqrt{{(- 4)}^{2}} + \sqrt{- 1} \cdot \sqrt{324})$

Step 2.1.9

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

$a + b i = 4 + 7 i - 2 (\sqrt{{(- 4)}^{2}} + i \cdot \sqrt{324})$

Step 2.1.10

Simplify each term.

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Step 2.1.10.1

Raise $- 4$ to the power of $2$ .

$a + b i = 4 + 7 i - 2 (\sqrt{16} + i \cdot \sqrt{324})$

Step 2.1.10.2

Rewrite $16$ as $4^{2}$ .

$a + b i = 4 + 7 i - 2 (\sqrt{4^{2}} + i \cdot \sqrt{324})$

Step 2.1.10.3

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

$a + b i = 4 + 7 i - 2 (4 + i \cdot \sqrt{324})$

Step 2.1.10.4

Rewrite $324$ as $18^{2}$ .

$a + b i = 4 + 7 i - 2 (4 + i \cdot \sqrt{18^{2}})$

Step 2.1.10.5

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

$a + b i = 4 + 7 i - 2 (4 + i \cdot 18)$

Step 2.1.10.6

Move $18$ to the left of $i$ .

$a + b i = 4 + 7 i - 2 (4 + 18 i)$

Step 2.1.11

Apply the distributive property.

$a + b i = 4 + 7 i - 2 \cdot 4 - 2 (18 i)$

Step 2.1.12

Multiply $- 2$ by $4$ .

$a + b i = 4 + 7 i - 8 - 2 (18 i)$

Step 2.1.13

Multiply $18$ by $- 2$ .

$a + b i = 4 + 7 i - 8 - 36 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $8$ from $4$ .

$a + b i = - 4 + 7 i - 36 i$

Step 2.2.2

Subtract $36 i$ from $7 i$ .

$a + b i = - 4 - 29 i$

Step 3

Subtract $b i$ from both sides of the equation.

$a = - 4 - 29 i - b i$