Algebra Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator and denominator of

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

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

2 + 1 i

$2 + 1 i$ to make the denominator real.

\frac{2 i}{2 + 1 i} \cdot \frac{2 - i}{2 - i} + \frac{5}{2 - i}

$\frac{2 i}{2 + 1 i} \cdot \frac{2 - i}{2 - i} + \frac{5}{2 - i}$

Step 1.2

Multiply.

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Step 1.2.1

Combine.

\frac{2 i (2 - i)}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Apply the distributive property.

\frac{2 i \cdot 2 + 2 i (- i)}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.2.2

Multiply

2

$2$ by

2

$2$ .

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

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

Step 1.2.2.3

Multiply

2 i (- i)

$2 i (- i)$ .

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Step 1.2.2.3.1

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 1.2.2.3.2

Raise

i

$i$ to the power of

1

$1$ .

\frac{4 i - 2 (i^{1} i)}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.2.3.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{4 i - 2 (i^{1} i^{1})}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.2.3.4

Use the power rule

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

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

\frac{4 i - 2 i^{1 + 1}}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.2.3.5

Add

1

$1$ and

1

$1$ .

\frac{4 i - 2 i^{2}}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

\frac{4 i - 2 i^{2}}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.2.4

Simplify each term.

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Step 1.2.2.4.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{4 i - 2 \cdot - 1}{(2 + 1 i) (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.2.4.2

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

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

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

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

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

Step 1.2.2.5

Reorder

4 i

$4 i$ and

2

$2$ .

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

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

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

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

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

Expand

(2 + 1 i) (2 - i)

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

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Step 1.2.3.1.1

Apply the distributive property.

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

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

Step 1.2.3.1.2

Apply the distributive property.

\frac{2 + 4 i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)} + \frac{5}{2 - i}

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

Step 1.2.3.1.3

Apply the distributive property.

\frac{2 + 4 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)} + \frac{5}{2 - i}

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

\frac{2 + 4 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)} + \frac{5}{2 - i}

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

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Multiply

2

$2$ by

2

$2$ .

\frac{2 + 4 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)} + \frac{5}{2 - i}

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

Step 1.2.3.2.2

Multiply

- 1

$- 1$ by

2

$2$ .

\frac{2 + 4 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)} + \frac{5}{2 - i}

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

Step 1.2.3.2.3

Multiply

2

$2$ by

1

$1$ .

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

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

Step 1.2.3.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 1.2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{2 + 4 i}{4 - 2 i + 2 i - (i^{1} i)} + \frac{5}{2 - i}

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

Step 1.2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{2 + 4 i}{4 - 2 i + 2 i - (i^{1} i^{1})} + \frac{5}{2 - i}

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

Step 1.2.3.2.7

Use the power rule

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

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

\frac{2 + 4 i}{4 - 2 i + 2 i - i^{1 + 1}} + \frac{5}{2 - i}

$\frac{2 + 4 i}{4 - 2 i + 2 i - i^{1 + 1}} + \frac{5}{2 - i}$

Step 1.2.3.2.8

Add

1

$1$ and

1

$1$ .

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

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

Step 1.2.3.2.9

Add

- 2 i

$- 2 i$ and

2 i

$2 i$ .

\frac{2 + 4 i}{4 + 0 - i^{2}} + \frac{5}{2 - i}

$\frac{2 + 4 i}{4 + 0 - i^{2}} + \frac{5}{2 - i}$

Step 1.2.3.2.10

Add

4

$4$ and

0

$0$ .

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

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

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

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

Step 1.2.3.3

Simplify each term.

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Step 1.2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{2 + 4 i}{4 - - 1} + \frac{5}{2 - i}

$\frac{2 + 4 i}{4 - - 1} + \frac{5}{2 - i}$

Step 1.2.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{2 + 4 i}{4 + 1} + \frac{5}{2 - i}

$\frac{2 + 4 i}{4 + 1} + \frac{5}{2 - i}$

\frac{2 + 4 i}{4 + 1} + \frac{5}{2 - i}

$\frac{2 + 4 i}{4 + 1} + \frac{5}{2 - i}$

Step 1.2.3.4

Add

4

$4$ and

1

$1$ .

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

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

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

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

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

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

Step 1.3

Split the fraction

\frac{2 + 4 i}{5}

$\frac{2 + 4 i}{5}$ into two fractions.

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

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

Step 1.4

Multiply the numerator and denominator of

\frac{5}{2 - i}

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

2 - i

$2 - i$ to make the denominator real.

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

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

Step 1.5

Multiply.

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Step 1.5.1

Combine.

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

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

Step 1.5.2

Simplify the numerator.

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Step 1.5.2.1

Apply the distributive property.

\frac{2}{5} + \frac{4 i}{5} + \frac{5 \cdot 2 + 5 i}{(2 - i) (2 + i)}

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

Step 1.5.2.2

Multiply

5

$5$ by

2

$2$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{(2 - i) (2 + i)}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{(2 - i) (2 + i)}$

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{(2 - i) (2 + i)}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{(2 - i) (2 + i)}$

Step 1.5.3

Simplify the denominator.

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Step 1.5.3.1

Expand

(2 - i) (2 + i)

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

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Step 1.5.3.1.1

Apply the distributive property.

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 (2 + i) - i (2 + i)}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 (2 + i) - i (2 + i)}$

Step 1.5.3.1.2

Apply the distributive property.

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 \cdot 2 + 2 i - i (2 + i)}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 \cdot 2 + 2 i - i (2 + i)}$

Step 1.5.3.1.3

Apply the distributive property.

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}$

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{2 \cdot 2 + 2 i - i \cdot 2 - i i}$

Step 1.5.3.2

Simplify.

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Step 1.5.3.2.1

Multiply

2

$2$ by

2

$2$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - i \cdot 2 - i i}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - i \cdot 2 - i i}$

Step 1.5.3.2.2

Multiply

2

$2$ by

- 1

$- 1$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - i i}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - i i}$

Step 1.5.3.2.3

Raise

i

$i$ to the power of

1

$1$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - (i^{1} i)}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - (i^{1} i)}$

Step 1.5.3.2.4

Raise

i

$i$ to the power of

1

$1$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - (i^{1} i^{1})}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - (i^{1} i^{1})}$

Step 1.5.3.2.5

Use the power rule

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

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

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - i^{1 + 1}}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - i^{1 + 1}}$

Step 1.5.3.2.6

Add

1

$1$ and

1

$1$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - i^{2}}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 2 i - 2 i - i^{2}}$

Step 1.5.3.2.7

Subtract

2 i

$2 i$ from

2 i

$2 i$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 0 - i^{2}}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 0 - i^{2}}$

Step 1.5.3.2.8

Add

4

$4$ and

0

$0$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 - i^{2}}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 - i^{2}}$

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 - i^{2}}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 - i^{2}}$

Step 1.5.3.3

Simplify each term.

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Step 1.5.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 - - 1}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 - - 1}$

Step 1.5.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 1}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 1}$

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 1}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{4 + 1}$

Step 1.5.3.4

Add

4

$4$ and

1

$1$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{5}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{5}$

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{5}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{5}$

\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{5}

$\frac{2}{5} + \frac{4 i}{5} + \frac{10 + 5 i}{5}$

Step 1.6

Cancel the common factor of

10 + 5 i

$10 + 5 i$ and

5

$5$ .

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Step 1.6.1

Factor

5

$5$ out of

10

$10$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{5 \cdot 2 + 5 i}{5}

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

Step 1.6.2

Factor

5

$5$ out of

5 i

$5 i$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{5 \cdot 2 + 5 (i)}{5}

$\frac{2}{5} + \frac{4 i}{5} + \frac{5 \cdot 2 + 5 (i)}{5}$

Step 1.6.3

Factor

5

$5$ out of

5 (2) + 5 (i)

$5 (2) + 5 (i)$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{5 (2 + i)}{5}

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

Step 1.6.4

Cancel the common factors.

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Step 1.6.4.1

Factor

5

$5$ out of

5

$5$ .

\frac{2}{5} + \frac{4 i}{5} + \frac{5 (2 + i)}{5 (1)}

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

Step 1.6.4.2

Cancel the common factor.

\frac{2}{5} + \frac{4 i}{5} + \frac{5 (2 + i)}{5 \cdot 1}

$\frac{2}{5} + \frac{4 i}{5} + \frac{5 (2 + i)}{5 \cdot 1}$

Step 1.6.4.3

Rewrite the expression.

\frac{2}{5} + \frac{4 i}{5} + \frac{2 + i}{1}

$\frac{2}{5} + \frac{4 i}{5} + \frac{2 + i}{1}$

Step 1.6.4.4

Divide

2 + i

$2 + i$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 2

To write

2

$2$ as a fraction with a common denominator, multiply by

\frac{5}{5}

$\frac{5}{5}$ .

\frac{2}{5} + 2 \cdot \frac{5}{5} + \frac{4 i}{5} + i

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

Step 3

Combine

2

$2$ and

\frac{5}{5}

$\frac{5}{5}$ .

\frac{2}{5} + \frac{2 \cdot 5}{5} + \frac{4 i}{5} + i

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

Step 4

Combine the numerators over the common denominator.

\frac{2 + 2 \cdot 5}{5} + \frac{4 i}{5} + i

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply

2

$2$ by

5

$5$ .

\frac{2 + 10}{5} + \frac{4 i}{5} + i

$\frac{2 + 10}{5} + \frac{4 i}{5} + i$

Step 5.2

Add

2

$2$ and

10

$10$ .

\frac{12}{5} + \frac{4 i}{5} + i

$\frac{12}{5} + \frac{4 i}{5} + i$

\frac{12}{5} + \frac{4 i}{5} + i

$\frac{12}{5} + \frac{4 i}{5} + i$

Step 6

To write

i

$i$ as a fraction with a common denominator, multiply by

\frac{5}{5}

$\frac{5}{5}$ .

\frac{12}{5} + \frac{4 i}{5} + i \cdot \frac{5}{5}

$\frac{12}{5} + \frac{4 i}{5} + i \cdot \frac{5}{5}$

Step 7

Combine

i

$i$ and

\frac{5}{5}

$\frac{5}{5}$ .

\frac{12}{5} + \frac{4 i}{5} + \frac{i \cdot 5}{5}

$\frac{12}{5} + \frac{4 i}{5} + \frac{i \cdot 5}{5}$

Step 8

Combine the numerators over the common denominator.

\frac{12}{5} + \frac{9 i}{5}

$\frac{12}{5} + \frac{9 i}{5}$