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 $a$ from both sides of the equation.

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

Step 4

Divide each term in $b i = - 4 - 29 i - a$ by $i$ and simplify.

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Step 4.1

Divide each term in $b i = - 4 - 29 i - a$ by $i$ .

$\frac{b i}{i} = \frac{- 4}{i} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $i$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{b i}{i} = \frac{- 4}{i} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 4}{i} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator and denominator of $\frac{- 4}{i}$ by the conjugate of $i$ to make the denominator real.

$b = \frac{- 4}{i} \cdot \frac{i}{i} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.2

Multiply.

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Step 4.3.1.2.1

Combine.

$b = \frac{- 4 i}{i i} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2

Simplify the denominator.

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Step 4.3.1.2.2.1

Raise $i$ to the power of $1$ .

$b = \frac{- 4 i}{i^{1} i} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.2

Raise $i$ to the power of $1$ .

$b = \frac{- 4 i}{i^{1} i^{1}} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$b = \frac{- 4 i}{i^{1 + 1}} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.4

Add $1$ and $1$ .

$b = \frac{- 4 i}{i^{2}} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.2.2.5

Rewrite $i^{2}$ as $- 1$ .

$b = \frac{- 4 i}{- 1} + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.3

Move the negative one from the denominator of $\frac{- 4 i}{- 1}$ .

$b = - 1 \cdot (- 4 i) + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.4

Rewrite $- 1 \cdot (- 4 i)$ as $- (- 4 i)$ .

$b = - (- 4 i) + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.5

Multiply $- 4$ by $- 1$ .

$b = 4 i + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.6

Cancel the common factor of $i$ .

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Step 4.3.1.6.1

Cancel the common factor.

$b = 4 i + \frac{- 29 i}{i} + \frac{- a}{i}$

Step 4.3.1.6.2

Divide $- 29$ by $1$ .

$b = 4 i - 29 + \frac{- a}{i}$

Step 4.3.1.7

Multiply the numerator and denominator of $\frac{- a}{i}$ by the conjugate of $i$ to make the denominator real.

$b = 4 i - 29 + \frac{- a}{i} \cdot \frac{i}{i}$

Step 4.3.1.8

Multiply.

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Step 4.3.1.8.1

Combine.

$b = 4 i - 29 + \frac{- a i}{i i}$

Step 4.3.1.8.2

Simplify the denominator.

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Step 4.3.1.8.2.1

Raise $i$ to the power of $1$ .

$b = 4 i - 29 + \frac{- a i}{i^{1} i}$

Step 4.3.1.8.2.2

Raise $i$ to the power of $1$ .

$b = 4 i - 29 + \frac{- a i}{i^{1} i^{1}}$

Step 4.3.1.8.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$b = 4 i - 29 + \frac{- a i}{i^{1 + 1}}$

Step 4.3.1.8.2.4

Add $1$ and $1$ .

$b = 4 i - 29 + \frac{- a i}{i^{2}}$

Step 4.3.1.8.2.5

Rewrite $i^{2}$ as $- 1$ .

$b = 4 i - 29 + \frac{- a i}{- 1}$

Step 4.3.1.9

Dividing two negative values results in a positive value.

$b = 4 i - 29 + \frac{a i}{1}$

Step 4.3.1.10

Divide $a i$ by $1$ .

$b = 4 i - 29 + a i$

Step 4.3.2

Reorder $4 i$ and $- 29$ .

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