Algebra Examples

$\frac{1}{2 + 5 i} - \frac{4 + 3 i}{3 - i}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator and denominator of $\frac{1}{2 + 5 i}$ by the conjugate of $2 + 5 i$ to make the denominator real.

$\frac{1}{2 + 5 i} \cdot \frac{2 - 5 i}{2 - 5 i} - \frac{4 + 3 i}{3 - i}$

Step 1.2

Multiply.

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Step 1.2.1

Combine.

$\frac{1 (2 - 5 i)}{(2 + 5 i) (2 - 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.2

Multiply $2 - 5 i$ by $1$ .

$\frac{2 - 5 i}{(2 + 5 i) (2 - 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3

Simplify the denominator.

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Step 1.2.3.1

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

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Step 1.2.3.1.1

Apply the distributive property.

$\frac{2 - 5 i}{2 (2 - 5 i) + 5 i (2 - 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.1.2

Apply the distributive property.

$\frac{2 - 5 i}{2 \cdot 2 + 2 (- 5 i) + 5 i (2 - 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.1.3

Apply the distributive property.

$\frac{2 - 5 i}{2 \cdot 2 + 2 (- 5 i) + 5 i \cdot 2 + 5 i (- 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Multiply $2$ by $2$ .

$\frac{2 - 5 i}{4 + 2 (- 5 i) + 5 i \cdot 2 + 5 i (- 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.2

Multiply $- 5$ by $2$ .

$\frac{2 - 5 i}{4 - 10 i + 5 i \cdot 2 + 5 i (- 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.3

Multiply $2$ by $5$ .

$\frac{2 - 5 i}{4 - 10 i + 10 i + 5 i (- 5 i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.4

Multiply $- 5$ by $5$ .

$\frac{2 - 5 i}{4 - 10 i + 10 i - 25 i i} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.5

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

$\frac{2 - 5 i}{4 - 10 i + 10 i - 25 (i^{1} i)} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.6

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

$\frac{2 - 5 i}{4 - 10 i + 10 i - 25 (i^{1} i^{1})} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.7

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

$\frac{2 - 5 i}{4 - 10 i + 10 i - 25 i^{1 + 1}} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.8

Add $1$ and $1$ .

$\frac{2 - 5 i}{4 - 10 i + 10 i - 25 i^{2}} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.9

Add $- 10 i$ and $10 i$ .

$\frac{2 - 5 i}{4 + 0 - 25 i^{2}} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.2.10

Add $4$ and $0$ .

$\frac{2 - 5 i}{4 - 25 i^{2}} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.3

Simplify each term.

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Step 1.2.3.3.1

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

$\frac{2 - 5 i}{4 - 25 \cdot - 1} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.3.2

Multiply $- 25$ by $- 1$ .

$\frac{2 - 5 i}{4 + 25} - \frac{4 + 3 i}{3 - i}$

Step 1.2.3.4

Add $4$ and $25$ .

$\frac{2 - 5 i}{29} - \frac{4 + 3 i}{3 - i}$

Step 1.3

Split the fraction $\frac{2 - 5 i}{29}$ into two fractions.

$\frac{2}{29} + \frac{- 5 i}{29} - \frac{4 + 3 i}{3 - i}$

Step 1.4

Move the negative in front of the fraction.

$\frac{2}{29} - \frac{5 i}{29} - \frac{4 + 3 i}{3 - i}$

Step 1.5

Multiply the numerator and denominator of $\frac{4 + 3 i}{3 - i}$ by the conjugate of $3 - i$ to make the denominator real.

$\frac{2}{29} - \frac{5 i}{29} - (\frac{4 + 3 i}{3 - i} \cdot \frac{3 + i}{3 + i})$

Step 1.6

Multiply.

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Step 1.6.1

Combine.

$\frac{2}{29} - \frac{5 i}{29} - \frac{(4 + 3 i) (3 + i)}{(3 - i) (3 + i)}$

Step 1.6.2

Simplify the numerator.

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Step 1.6.2.1

Expand $(4 + 3 i) (3 + i)$ using the FOIL Method.

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Step 1.6.2.1.1

Apply the distributive property.

$\frac{2}{29} - \frac{5 i}{29} - \frac{4 (3 + i) + 3 i (3 + i)}{(3 - i) (3 + i)}$

Step 1.6.2.1.2

Apply the distributive property.

$\frac{2}{29} - \frac{5 i}{29} - \frac{4 \cdot 3 + 4 i + 3 i (3 + i)}{(3 - i) (3 + i)}$

Step 1.6.2.1.3

Apply the distributive property.

$\frac{2}{29} - \frac{5 i}{29} - \frac{4 \cdot 3 + 4 i + 3 i \cdot 3 + 3 i i}{(3 - i) (3 + i)}$

Step 1.6.2.2

Simplify and combine like terms.

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Step 1.6.2.2.1

Simplify each term.

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Step 1.6.2.2.1.1

Multiply $4$ by $3$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 3 i \cdot 3 + 3 i i}{(3 - i) (3 + i)}$

Step 1.6.2.2.1.2

Multiply $3$ by $3$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 9 i + 3 i i}{(3 - i) (3 + i)}$

Step 1.6.2.2.1.3

Multiply $3 i i$ .

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Step 1.6.2.2.1.3.1

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 9 i + 3 (i^{1} i)}{(3 - i) (3 + i)}$

Step 1.6.2.2.1.3.2

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 9 i + 3 (i^{1} i^{1})}{(3 - i) (3 + i)}$

Step 1.6.2.2.1.3.3

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 9 i + 3 i^{1 + 1}}{(3 - i) (3 + i)}$

Step 1.6.2.2.1.3.4

Add $1$ and $1$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 9 i + 3 i^{2}}{(3 - i) (3 + i)}$

Step 1.6.2.2.1.4

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 9 i + 3 \cdot - 1}{(3 - i) (3 + i)}$

Step 1.6.2.2.1.5

Multiply $3$ by $- 1$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{12 + 4 i + 9 i - 3}{(3 - i) (3 + i)}$

Step 1.6.2.2.2

Subtract $3$ from $12$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 4 i + 9 i}{(3 - i) (3 + i)}$

Step 1.6.2.2.3

Add $4 i$ and $9 i$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{(3 - i) (3 + i)}$

Step 1.6.3

Simplify the denominator.

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Step 1.6.3.1

Expand $(3 - i) (3 + i)$ using the FOIL Method.

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Step 1.6.3.1.1

Apply the distributive property.

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{3 (3 + i) - i (3 + i)}$

Step 1.6.3.1.2

Apply the distributive property.

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{3 \cdot 3 + 3 i - i (3 + i)}$

Step 1.6.3.1.3

Apply the distributive property.

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{3 \cdot 3 + 3 i - i \cdot 3 - i i}$

Step 1.6.3.2

Simplify.

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Step 1.6.3.2.1

Multiply $3$ by $3$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 3 i - i \cdot 3 - i i}$

Step 1.6.3.2.2

Multiply $3$ by $- 1$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 3 i - 3 i - i i}$

Step 1.6.3.2.3

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 3 i - 3 i - (i^{1} i)}$

Step 1.6.3.2.4

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 3 i - 3 i - (i^{1} i^{1})}$

Step 1.6.3.2.5

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 3 i - 3 i - i^{1 + 1}}$

Step 1.6.3.2.6

Add $1$ and $1$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 3 i - 3 i - i^{2}}$

Step 1.6.3.2.7

Subtract $3 i$ from $3 i$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 0 - i^{2}}$

Step 1.6.3.2.8

Add $9$ and $0$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 - i^{2}}$

Step 1.6.3.3

Simplify each term.

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Step 1.6.3.3.1

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

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 - - 1}$

Step 1.6.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{9 + 1}$

Step 1.6.3.4

Add $9$ and $1$ .

$\frac{2}{29} - \frac{5 i}{29} - \frac{9 + 13 i}{10}$

Step 1.7

Split the fraction $\frac{9 + 13 i}{10}$ into two fractions.

$\frac{2}{29} - \frac{5 i}{29} - (\frac{9}{10} + \frac{13 i}{10})$

Step 1.8

Apply the distributive property.

$\frac{2}{29} - \frac{5 i}{29} - \frac{9}{10} - \frac{13 i}{10}$

Step 2

To write $\frac{2}{29}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{2}{29} \cdot \frac{10}{10} - \frac{9}{10} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 3

To write $- \frac{9}{10}$ as a fraction with a common denominator, multiply by $\frac{29}{29}$ .

$\frac{2}{29} \cdot \frac{10}{10} - \frac{9}{10} \cdot \frac{29}{29} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 4

Write each expression with a common denominator of $290$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{29}$ by $\frac{10}{10}$ .

$\frac{2 \cdot 10}{29 \cdot 10} - \frac{9}{10} \cdot \frac{29}{29} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 4.2

Multiply $29$ by $10$ .

$\frac{2 \cdot 10}{290} - \frac{9}{10} \cdot \frac{29}{29} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 4.3

Multiply $\frac{9}{10}$ by $\frac{29}{29}$ .

$\frac{2 \cdot 10}{290} - \frac{9 \cdot 29}{10 \cdot 29} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 4.4

Multiply $10$ by $29$ .

$\frac{2 \cdot 10}{290} - \frac{9 \cdot 29}{290} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 \cdot 10 - 9 \cdot 29}{290} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $2$ by $10$ .

$\frac{20 - 9 \cdot 29}{290} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 6.2

Multiply $- 9$ by $29$ .

$\frac{20 - 261}{290} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 6.3

Subtract $261$ from $20$ .

$\frac{- 241}{290} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 7

Move the negative in front of the fraction.

$- \frac{241}{290} - \frac{5 i}{29} - \frac{13 i}{10}$

Step 8

To write $- \frac{5 i}{29}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$- \frac{241}{290} - \frac{5 i}{29} \cdot \frac{10}{10} - \frac{13 i}{10}$

Step 9

To write $- \frac{13 i}{10}$ as a fraction with a common denominator, multiply by $\frac{29}{29}$ .

$- \frac{241}{290} - \frac{5 i}{29} \cdot \frac{10}{10} - \frac{13 i}{10} \cdot \frac{29}{29}$

Step 10

Write each expression with a common denominator of $290$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{5 i}{29}$ by $\frac{10}{10}$ .

$- \frac{241}{290} - \frac{5 i \cdot 10}{29 \cdot 10} - \frac{13 i}{10} \cdot \frac{29}{29}$

Step 10.2

Multiply $29$ by $10$ .

$- \frac{241}{290} - \frac{5 i \cdot 10}{290} - \frac{13 i}{10} \cdot \frac{29}{29}$

Step 10.3

Multiply $\frac{13 i}{10}$ by $\frac{29}{29}$ .

$- \frac{241}{290} - \frac{5 i \cdot 10}{290} - \frac{13 i \cdot 29}{10 \cdot 29}$

Step 10.4

Multiply $10$ by $29$ .

$- \frac{241}{290} - \frac{5 i \cdot 10}{290} - \frac{13 i \cdot 29}{290}$

Step 11

Combine the numerators over the common denominator.

$- \frac{241}{290} + \frac{- 427 i}{290}$

Step 12

Move the negative in front of the fraction.

$- \frac{241}{290} - \frac{427 i}{290}$