Finite Math Examples

$\frac{\sqrt{7} + 5 i}{\sqrt{5} + 2 i}$

Step 1

Multiply $\frac{\sqrt{7} + 5 i}{\sqrt{5} + 2 i}$ by $\frac{\sqrt{5} - 2 i}{\sqrt{5} - 2 i}$ .

$\frac{\sqrt{7} + 5 i}{\sqrt{5} + 2 i} \cdot \frac{\sqrt{5} - 2 i}{\sqrt{5} - 2 i}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{7} + 5 i}{\sqrt{5} + 2 i}$ by $\frac{\sqrt{5} - 2 i}{\sqrt{5} - 2 i}$ .

$\frac{(\sqrt{7} + 5 i) (\sqrt{5} - 2 i)}{(\sqrt{5} + 2 i) (\sqrt{5} - 2 i)}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{7} + 5 i) (\sqrt{5} - 2 i)}{{\sqrt{5}}^{2} + \sqrt{5} (- 2 i) + 2 i \sqrt{5} - 4 i^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{7} + 5 i) (\sqrt{5} - 2 i)}{9}$

Step 3

Expand $(\sqrt{7} + 5 i) (\sqrt{5} - 2 i)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{7} (\sqrt{5} - 2 i) + 5 i (\sqrt{5} - 2 i)}{9}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{5} + \sqrt{7} (- 2 i) + 5 i (\sqrt{5} - 2 i)}{9}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{5} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} + 5 i (- 2 i)}{9}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 5} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} + 5 i (- 2 i)}{9}$

Step 4.1.2

Multiply $7$ by $5$ .

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} + 5 i (- 2 i)}{9}$

Step 4.1.3

Multiply $5 i (- 2 i)$ .

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Step 4.1.3.1

Multiply $- 2$ by $5$ .

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} - 10 i i}{9}$

Step 4.1.3.2

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

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} - 10 (i^{1} i)}{9}$

Step 4.1.3.3

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

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} - 10 (i^{1} i^{1})}{9}$

Step 4.1.3.4

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

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} - 10 i^{1 + 1}}{9}$

Step 4.1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} - 10 i^{2}}{9}$

Step 4.1.4

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

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} - 10 \cdot - 1}{9}$

Step 4.1.5

Multiply $- 10$ by $- 1$ .

$\frac{\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5} + 10}{9}$

Step 4.2

Reorder $\sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5}$ and $10$ .

$\frac{10 + \sqrt{35} + \sqrt{7} (- 2 i) + 5 i \sqrt{5}}{9}$