Algebra Examples

$\frac{\sqrt{- 9}}{(4 - 7 i) - (6 - 6 i)}$

Step 1

Pull out imaginary unit $i$ .

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Step 1.1

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{\sqrt{- 1 (9)}}{(4 - 7 i) - (6 - 6 i)}$

Step 1.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$\frac{\sqrt{- 1} \cdot \sqrt{9}}{(4 - 7 i) - (6 - 6 i)}$

Step 1.3

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

$\frac{i \cdot \sqrt{9}}{(4 - 7 i) - (6 - 6 i)}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{i \sqrt{3^{2}}}{4 - 7 i - (6 - 6 i)}$

Step 2.2

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

$\frac{i \cdot 3}{4 - 7 i - (6 - 6 i)}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the distributive property.

$\frac{i \cdot 3}{4 - 7 i - 1 \cdot 6 - (- 6 i)}$

Step 3.2

Multiply $- 1$ by $6$ .

$\frac{i \cdot 3}{4 - 7 i - 6 - (- 6 i)}$

Step 3.3

Multiply $- 6$ by $- 1$ .

$\frac{i \cdot 3}{4 - 7 i - 6 + 6 i}$

Step 3.4

Subtract $6$ from $4$ .

$\frac{i \cdot 3}{- 2 - 7 i + 6 i}$

Step 3.5

Add $- 7 i$ and $6 i$ .

$\frac{i \cdot 3}{- 2 - i}$

Step 4

Move $3$ to the left of $i$ .

$\frac{3 i}{- 2 - i}$

Step 5

Multiply the numerator and denominator of $\frac{3 i}{- 2 - i}$ by the conjugate of $- 2 - i$ to make the denominator real.

$\frac{3 i}{- 2 - i} \cdot \frac{- 2 + i}{- 2 + i}$

Step 6

Multiply.

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Step 6.1

Combine.

$\frac{3 i (- 2 + i)}{(- 2 - i) (- 2 + i)}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Apply the distributive property.

$\frac{3 i \cdot - 2 + 3 i i}{(- 2 - i) (- 2 + i)}$

Step 6.2.2

Multiply $- 2$ by $3$ .

$\frac{- 6 i + 3 i i}{(- 2 - i) (- 2 + i)}$

Step 6.2.3

Multiply $3 i i$ .

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Step 6.2.3.1

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

$\frac{- 6 i + 3 (i^{1} i)}{(- 2 - i) (- 2 + i)}$

Step 6.2.3.2

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

$\frac{- 6 i + 3 (i^{1} i^{1})}{(- 2 - i) (- 2 + i)}$

Step 6.2.3.3

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

$\frac{- 6 i + 3 i^{1 + 1}}{(- 2 - i) (- 2 + i)}$

Step 6.2.3.4

Add $1$ and $1$ .

$\frac{- 6 i + 3 i^{2}}{(- 2 - i) (- 2 + i)}$

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

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

$\frac{- 6 i + 3 \cdot - 1}{(- 2 - i) (- 2 + i)}$

Step 6.2.4.2

Multiply $3$ by $- 1$ .

$\frac{- 6 i - 3}{(- 2 - i) (- 2 + i)}$

Step 6.2.5

Reorder $- 6 i$ and $- 3$ .

$\frac{- 3 - 6 i}{(- 2 - i) (- 2 + i)}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Expand $(- 2 - i) (- 2 + i)$ using the FOIL Method.

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Step 6.3.1.1

Apply the distributive property.

$\frac{- 3 - 6 i}{- 2 (- 2 + i) - i (- 2 + i)}$

Step 6.3.1.2

Apply the distributive property.

$\frac{- 3 - 6 i}{- 2 \cdot - 2 - 2 i - i (- 2 + i)}$

Step 6.3.1.3

Apply the distributive property.

$\frac{- 3 - 6 i}{- 2 \cdot - 2 - 2 i - i \cdot - 2 - i i}$

Step 6.3.2

Simplify.

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Step 6.3.2.1

Multiply $- 2$ by $- 2$ .

$\frac{- 3 - 6 i}{4 - 2 i - i \cdot - 2 - i i}$

Step 6.3.2.2

Multiply $- 2$ by $- 1$ .

$\frac{- 3 - 6 i}{4 - 2 i + 2 i - i i}$

Step 6.3.2.3

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

$\frac{- 3 - 6 i}{4 - 2 i + 2 i - (i^{1} i)}$

Step 6.3.2.4

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

$\frac{- 3 - 6 i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

Step 6.3.2.5

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

$\frac{- 3 - 6 i}{4 - 2 i + 2 i - i^{1 + 1}}$

Step 6.3.2.6

Add $1$ and $1$ .

$\frac{- 3 - 6 i}{4 - 2 i + 2 i - i^{2}}$

Step 6.3.2.7

Add $- 2 i$ and $2 i$ .

$\frac{- 3 - 6 i}{4 + 0 - i^{2}}$

Step 6.3.2.8

Add $4$ and $0$ .

$\frac{- 3 - 6 i}{4 - i^{2}}$

Step 6.3.3

Simplify each term.

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Step 6.3.3.1

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

$\frac{- 3 - 6 i}{4 - - 1}$

Step 6.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{- 3 - 6 i}{4 + 1}$

Step 6.3.4

Add $4$ and $1$ .

$\frac{- 3 - 6 i}{5}$

Step 7

Split the fraction $\frac{- 3 - 6 i}{5}$ into two fractions.

$\frac{- 3}{5} + \frac{- 6 i}{5}$

Step 8

Simplify each term.

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Step 8.1

Move the negative in front of the fraction.

$- \frac{3}{5} + \frac{- 6 i}{5}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{3}{5} - \frac{6 i}{5}$