Basic Math Examples

$\frac{10}{10 + 5 i}$

Step 1

Cancel the common factor of $10$ and $10 + 5 i$ .

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Step 1.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 2}{10 + 5 i}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 2}{5 (2) + 5 i}$

Step 1.2.2

Factor $5$ out of $5 i$ .

$\frac{5 \cdot 2}{5 (2) + 5 (i)}$

Step 1.2.3

Factor $5$ out of $5 (2) + 5 (i)$ .

$\frac{5 \cdot 2}{5 (2 + i)}$

Step 1.2.4

Cancel the common factor.

$\frac{5 \cdot 2}{5 (2 + i)}$

Step 1.2.5

Rewrite the expression.

$\frac{2}{2 + i}$

Step 2

Multiply the numerator and denominator of $\frac{2}{2 + 1 i}$ by the conjugate of $2 + 1 i$ to make the denominator real.

$\frac{2}{2 + 1 i} \cdot \frac{2 - i}{2 - i}$

Step 3

Multiply.

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Step 3.1

Combine.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 3.2.3

Multiply $- 1$ by $2$ .

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

Expand $(2 + 1 i) (2 - i)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

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

Step 3.3.1.2

Apply the distributive property.

$\frac{4 - 2 i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)}$

Step 3.3.1.3

Apply the distributive property.

$\frac{4 - 2 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $2$ by $2$ .

$\frac{4 - 2 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 3.3.2.2

Multiply $- 1$ by $2$ .

$\frac{4 - 2 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)}$

Step 3.3.2.3

Multiply $2$ by $1$ .

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

Step 3.3.2.4

Multiply $- 1$ by $1$ .

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

Step 3.3.2.5

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

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

Step 3.3.2.6

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

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

Step 3.3.2.7

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

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

Step 3.3.2.8

Add $1$ and $1$ .

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

Step 3.3.2.9

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

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

Step 3.3.2.10

Add $4$ and $0$ .

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

Step 3.3.3

Simplify each term.

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Step 3.3.3.1

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

$\frac{4 - 2 i}{4 - - 1}$

Step 3.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{4 - 2 i}{4 + 1}$

Step 3.3.4

Add $4$ and $1$ .

$\frac{4 - 2 i}{5}$

Step 4

Split the fraction $\frac{4 - 2 i}{5}$ into two fractions.

$\frac{4}{5} + \frac{- 2 i}{5}$

Step 5

Move the negative in front of the fraction.

$\frac{4}{5} - \frac{2 i}{5}$