Basic Math Examples

\frac{6 + 5 i}{2 - i}

$\frac{6 + 5 i}{2 - i}$

Step 1

Multiply the numerator and denominator of

\frac{6 + 5 i}{2 - i}

$\frac{6 + 5 i}{2 - i}$ by the conjugate of

2 - i

$2 - i$ to make the denominator real.

\frac{6 + 5 i}{2 - i} \cdot \frac{2 + i}{2 + i}

$\frac{6 + 5 i}{2 - i} \cdot \frac{2 + i}{2 + i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{(6 + 5 i) (2 + i)}{(2 - i) (2 + i)}

$\frac{(6 + 5 i) (2 + i)}{(2 - i) (2 + i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Expand

(6 + 5 i) (2 + i)

$(6 + 5 i) (2 + i)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

\frac{6 (2 + i) + 5 i (2 + i)}{(2 - i) (2 + i)}

$\frac{6 (2 + i) + 5 i (2 + i)}{(2 - i) (2 + i)}$

Step 2.2.1.2

Apply the distributive property.

\frac{6 \cdot 2 + 6 i + 5 i (2 + i)}{(2 - i) (2 + i)}

$\frac{6 \cdot 2 + 6 i + 5 i (2 + i)}{(2 - i) (2 + i)}$

Step 2.2.1.3

Apply the distributive property.

\frac{6 \cdot 2 + 6 i + 5 i \cdot 2 + 5 i i}{(2 - i) (2 + i)}

$\frac{6 \cdot 2 + 6 i + 5 i \cdot 2 + 5 i i}{(2 - i) (2 + i)}$

\frac{6 \cdot 2 + 6 i + 5 i \cdot 2 + 5 i i}{(2 - i) (2 + i)}

$\frac{6 \cdot 2 + 6 i + 5 i \cdot 2 + 5 i i}{(2 - i) (2 + i)}$

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

6

$6$ by

2

$2$ .

\frac{12 + 6 i + 5 i \cdot 2 + 5 i i}{(2 - i) (2 + i)}

$\frac{12 + 6 i + 5 i \cdot 2 + 5 i i}{(2 - i) (2 + i)}$

Step 2.2.2.1.2

Multiply

2

$2$ by

5

$5$ .

\frac{12 + 6 i + 10 i + 5 i i}{(2 - i) (2 + i)}

$\frac{12 + 6 i + 10 i + 5 i i}{(2 - i) (2 + i)}$

Step 2.2.2.1.3

Multiply

5 i i

$5 i i$ .

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Step 2.2.2.1.3.1

Raise

i

$i$ to the power of

1

$1$ .

\frac{12 + 6 i + 10 i + 5 (i^{1} i)}{(2 - i) (2 + i)}

$\frac{12 + 6 i + 10 i + 5 (i^{1} i)}{(2 - i) (2 + i)}$

Step 2.2.2.1.3.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{12 + 6 i + 10 i + 5 (i^{1} i^{1})}{(2 - i) (2 + i)}

$\frac{12 + 6 i + 10 i + 5 (i^{1} i^{1})}{(2 - i) (2 + i)}$

Step 2.2.2.1.3.3

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{12 + 6 i + 10 i + 5 i^{1 + 1}}{(2 - i) (2 + i)}

$\frac{12 + 6 i + 10 i + 5 i^{1 + 1}}{(2 - i) (2 + i)}$

Step 2.2.2.1.3.4

Add

1

$1$ and

1

$1$ .

\frac{12 + 6 i + 10 i + 5 i^{2}}{(2 - i) (2 + i)}

$\frac{12 + 6 i + 10 i + 5 i^{2}}{(2 - i) (2 + i)}$

Step 2.2.2.1.4

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{12 + 6 i + 10 i + 5 \cdot - 1}{(2 - i) (2 + i)}$

Step 2.2.2.1.5

Multiply

$5$ by

$- 1$ .

$\frac{12 + 6 i + 10 i - 5}{(2 - i) (2 + i)}$

Step 2.2.2.2

Subtract

$5$ from

$12$ .

$\frac{7 + 6 i + 10 i}{(2 - i) (2 + i)}$

Step 2.2.2.3

Add

$6 i$ and

$10 i$ .

$\frac{7 + 16 i}{(2 - i) (2 + i)}$

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.

$\frac{7 + 16 i}{2 (2 + i) - i (2 + i)}$

Step 2.3.1.2

Apply the distributive property.

$\frac{7 + 16 i}{2 \cdot 2 + 2 i - i (2 + i)}$

Step 2.3.1.3

Apply the distributive property.

$\frac{7 + 16 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

$2$ by

$2$ .

$\frac{7 + 16 i}{4 + 2 i - i \cdot 2 - i i}$

Step 2.3.2.2

Multiply

$2$ by

$- 1$ .

$\frac{7 + 16 i}{4 + 2 i - 2 i - i i}$

Step 2.3.2.3

Raise

$i$ to the power of

$1$ .

$\frac{7 + 16 i}{4 + 2 i - 2 i - (i^{1} i)}$

Step 2.3.2.4

Raise

$i$ to the power of

$1$ .

$\frac{7 + 16 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

Step 2.3.2.5

Use the power rule

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

$\frac{7 + 16 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Step 2.3.2.6

Add

$1$ and

$1$ .

$\frac{7 + 16 i}{4 + 2 i - 2 i - i^{2}}$

Step 2.3.2.7

Subtract

$2 i$ from

$2 i$ .

$\frac{7 + 16 i}{4 + 0 - i^{2}}$

Step 2.3.2.8

Add

$4$ and

$0$ .

$\frac{7 + 16 i}{4 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

$i^{2}$ as

$- 1$ .

$\frac{7 + 16 i}{4 - - 1}$

Step 2.3.3.2

Multiply

$- 1$ by

$- 1$ .

$\frac{7 + 16 i}{4 + 1}$

Step 2.3.4

Add

$4$ and

$1$ .

$\frac{7 + 16 i}{5}$

Step 3

Split the fraction

$\frac{7 + 16 i}{5}$ into two fractions.

$\frac{7}{5} + \frac{16 i}{5}$