Algebra Examples

$4 + 3 i$ , $2 + \sqrt{5}$

Step 1

Rewrite the division as a fraction.

$\frac{4 + 3 i}{2 + \sqrt{5}}$

Step 2

Multiply $\frac{4 + 3 i}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

$\frac{4 + 3 i}{2 + \sqrt{5}} \cdot \frac{2 - \sqrt{5}}{2 - \sqrt{5}}$

Step 3

Combine fractions.

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Step 3.1

Multiply $\frac{4 + 3 i}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

$\frac{(4 + 3 i) (2 - \sqrt{5})}{(2 + \sqrt{5}) (2 - \sqrt{5})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(4 + 3 i) (2 - \sqrt{5})}{4 - 2 \sqrt{5} + \sqrt{5} \cdot 2 - {\sqrt{5}}^{2}}$

Step 3.3

Simplify.

$\frac{(4 + 3 i) (2 - \sqrt{5})}{- 1}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Move the negative one from the denominator of $\frac{(4 + 3 i) (2 - \sqrt{5})}{- 1}$ .

$- 1 \cdot ((4 + 3 i) (2 - \sqrt{5}))$

Step 3.4.2

Rewrite $- 1 \cdot ((4 + 3 i) (2 - \sqrt{5}))$ as $- ((4 + 3 i) (2 - \sqrt{5}))$ .

$- ((4 + 3 i) (2 - \sqrt{5}))$

Step 4

Expand $(4 + 3 i) (2 - \sqrt{5})$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$- (4 (2 - \sqrt{5}) + 3 i (2 - \sqrt{5}))$

Step 4.2

Apply the distributive property.

$- (4 \cdot 2 + 4 (- \sqrt{5}) + 3 i (2 - \sqrt{5}))$

Step 4.3

Apply the distributive property.

$- (4 \cdot 2 + 4 (- \sqrt{5}) + 3 i \cdot 2 + 3 i (- \sqrt{5}))$

Step 5

Simplify terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $4$ by $2$ .

$- (8 + 4 (- \sqrt{5}) + 3 i \cdot 2 + 3 i (- \sqrt{5}))$

Step 5.1.2

Multiply $- 1$ by $4$ .

$- (8 - 4 \sqrt{5} + 3 i \cdot 2 + 3 i (- \sqrt{5}))$

Step 5.1.3

Multiply $2$ by $3$ .

$- (8 - 4 \sqrt{5} + 6 i + 3 i (- \sqrt{5}))$

Step 5.1.4

Multiply $- 1$ by $3$ .

$- (8 - 4 \sqrt{5} + 6 i - 3 i \sqrt{5})$

Step 5.2

Apply the distributive property.

$- 1 \cdot 8 - (- 4 \sqrt{5}) - (6 i) - (- 3 i \sqrt{5})$

Step 6

Simplify.

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Step 6.1

Multiply $- 1$ by $8$ .

$- 8 - (- 4 \sqrt{5}) - (6 i) - (- 3 i \sqrt{5})$

Step 6.2

Multiply $- 4$ by $- 1$ .

$- 8 + 4 \sqrt{5} - (6 i) - (- 3 i \sqrt{5})$

Step 6.3

Multiply $6$ by $- 1$ .

$- 8 + 4 \sqrt{5} - 6 i - (- 3 i \sqrt{5})$

Step 6.4

Multiply $- 3$ by $- 1$ .

$- 8 + 4 \sqrt{5} - 6 i + 3 (i \sqrt{5})$

$- 8 + 4 \sqrt{5} - 6 i + 3 i \sqrt{5}$