Basic Math Examples

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

Step 1

Multiply the numerator and denominator of

$\frac{1 + i}{2 - i}$ by the conjugate of

$2 - i$ to make the denominator real.

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply

$2$ by

$1$ .

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

Step 2.2.2.1.2

Multiply

$i$ by

$1$ .

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

Step 2.2.2.1.3

Move

$2$ to the left of

$i$ .

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

Step 2.2.2.1.4

Multiply

$i i$ .

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Step 2.2.2.1.4.1

Raise

$i$ to the power of

$1$ .

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

Step 2.2.2.1.4.2

Raise

$i$ to the power of

$1$ .

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

Step 2.2.2.1.4.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.2.1.4.4

Add

$1$ and

$1$ .

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

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

Step 2.2.2.1.5

Rewrite

$i^{2}$ as

$- 1$ .

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

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

Step 2.2.2.2

Subtract

$1$ from

$2$ .

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

Step 2.2.2.3

Add

$i$ and

$2 i$ .

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

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

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

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

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$2$ by

$2$ .

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

Step 2.3.2.2

Multiply

$2$ by

$- 1$ .

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

Step 2.3.2.3

Raise

$i$ to the power of

$1$ .

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

Step 2.3.2.4

Raise

$i$ to the power of

$1$ .

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

Step 2.3.2.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.2.6

Add

$1$ and

$1$ .

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

Step 2.3.2.7

Subtract

$2 i$ from

$2 i$ .

$\frac{1 + 3 i}{4 + 0 - i^{2}}$

Step 2.3.2.8

Add

$4$ and

$0$ .

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

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

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

Step 2.3.3.2

Multiply

$- 1$ by

$- 1$ .

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

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

Step 2.3.4

Add

$4$ and

$1$ .

$\frac{1 + 3 i}{5}$

$\frac{1 + 3 i}{5}$

$\frac{1 + 3 i}{5}$

Step 3

Split the fraction

$\frac{1 + 3 i}{5}$ into two fractions.

$\frac{1}{5} + \frac{3 i}{5}$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$