Calculus Examples

$z = - \frac{4}{\sqrt{3} - i}$

Step 1

Multiply the numerator and denominator of $\frac{4}{1.7320508 - i}$ by the conjugate of $1.7320508 - i$ to make the denominator real.

$z = - (\frac{4}{1.7320508 - i} \cdot \frac{1.7320508 + i}{1.7320508 + i})$

Step 2

Multiply.

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Step 2.1

Combine.

$z = - \frac{4 (1.7320508 + i)}{(1.7320508 - i) (1.7320508 + i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$z = - \frac{4 \cdot 1.7320508 + 4 i}{(1.7320508 - i) (1.7320508 + i)}$

Step 2.2.2

Multiply $4$ by $1.7320508$ .

$z = - \frac{6.92820323 + 4 i}{(1.7320508 - i) (1.7320508 + i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(1.7320508 - i) (1.7320508 + i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

$z = - \frac{6.92820323 + 4 i}{1.7320508 (1.7320508 + i) - i (1.7320508 + i)}$

Step 2.3.1.2

Apply the distributive property.

$z = - \frac{6.92820323 + 4 i}{1.7320508 \cdot 1.7320508 + 1.7320508 i - i (1.7320508 + i)}$

Step 2.3.1.3

Apply the distributive property.

$z = - \frac{6.92820323 + 4 i}{1.7320508 \cdot 1.7320508 + 1.7320508 i - i \cdot 1.7320508 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $1.7320508$ by $1.7320508$ .

$z = - \frac{6.92820323 + 4 i}{3 + 1.7320508 i - i \cdot 1.7320508 - i i}$

Step 2.3.2.2

Multiply $1.7320508$ by $- 1$ .

$z = - \frac{6.92820323 + 4 i}{3 + 1.7320508 i - 1.7320508 i - i i}$

Step 2.3.2.3

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

$z = - \frac{6.92820323 + 4 i}{3 + 1.7320508 i - 1.7320508 i - (i^{1} i)}$

Step 2.3.2.4

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

$z = - \frac{6.92820323 + 4 i}{3 + 1.7320508 i - 1.7320508 i - (i^{1} i^{1})}$

Step 2.3.2.5

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

$z = - \frac{6.92820323 + 4 i}{3 + 1.7320508 i - 1.7320508 i - i^{1 + 1}}$

Step 2.3.2.6

Add $1$ and $1$ .

$z = - \frac{6.92820323 + 4 i}{3 + 1.7320508 i - 1.7320508 i - i^{2}}$

Step 2.3.2.7

Subtract $1.7320508 i$ from $1.7320508 i$ .

$z = - \frac{6.92820323 + 4 i}{3 + 0 i - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Multiply $0$ by $i$ .

$z = - \frac{6.92820323 + 4 i}{3 + 0 - i^{2}}$

Step 2.3.3.2

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

$z = - \frac{6.92820323 + 4 i}{3 + 0 - - 1}$

Step 2.3.3.3

Multiply $- 1$ by $- 1$ .

$z = - \frac{6.92820323 + 4 i}{3 + 0 + 1}$

Step 2.3.4

Add $3$ and $0$ .

$z = - \frac{6.92820323 + 4 i}{3 + 1}$

Step 2.3.5

Add $3$ and $1$ .

$z = - \frac{6.92820323 + 4 i}{4}$

Step 3

Rewrite $6.92820323$ as $1 (6.92820323)$ .

$z = - \frac{1 (6.92820323) + 4 i}{4}$

Step 4

Factor $1$ out of $4 i$ .

$z = - \frac{1 (6.92820323) + 1 (4 i)}{4}$

Step 5

Factor $1$ out of $1 (6.92820323) + 1 (4 i)$ .

$z = - \frac{1 (6.92820323 + 4 i)}{4}$

Step 6

Factor $4$ out of $4$ .

$z = - \frac{1 (6.92820323 + 4 i)}{4 (1)}$

Step 7

Separate fractions.

$z = - (\frac{1}{4} \cdot \frac{6.92820323 + 4 i}{1})$

Step 8

Simplify the expression.

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Step 8.1

Divide $1$ by $4$ .

$z = - (0.25 \frac{6.92820323 + 4 i}{1})$

Step 8.2

Divide $6.92820323 + 4 i$ by $1$ .

$z = - (0.25 (6.92820323 + 4 i))$

Step 9

Apply the distributive property.

$z = - (0.25 \cdot 6.92820323 + 0.25 (4 i))$

Step 10

Multiply.

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Step 10.1

Multiply $0.25$ by $6.92820323$ .

$z = - (1.7320508 + 0.25 (4 i))$

Step 10.2

Multiply $4$ by $0.25$ .

$z = - (1.7320508 + 1 i)$

Step 11

Apply the distributive property.

$z = - 1 \cdot 1.7320508 - (1 i)$

Step 12

Multiply.

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Step 12.1

Multiply $- 1$ by $1.7320508$ .

$z = - 1.7320508 - (1 i)$

Step 12.2

Multiply $1$ by $- 1$ .

$z = - 1.7320508 - 1 i$