Simplify 5/(3-i)+2/(3+i)

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

Step 1

Simplify each term.

Multiply the numerator and denominator of $\frac{5}{3 - i}$ by the conjugate of $3 - i$ to make the denominator real.

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $3$ .

$\frac{15 + 5 i}{(3 - i) (3 + i)} + \frac{2}{3 + i}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{15 + 5 i}{3 (3 + i) - i (3 + i)} + \frac{2}{3 + i}$

Apply the distributive property.

$\frac{15 + 5 i}{3 \cdot 3 + 3 i - i (3 + i)} + \frac{2}{3 + i}$

Apply the distributive property.

$\frac{15 + 5 i}{3 \cdot 3 + 3 i - i \cdot 3 - i i} + \frac{2}{3 + i}$

Simplify.

Multiply $3$ by $3$ .

$\frac{15 + 5 i}{9 + 3 i - i \cdot 3 - i i} + \frac{2}{3 + i}$

Multiply $3$ by $- 1$ .

$\frac{15 + 5 i}{9 + 3 i - 3 i - i i} + \frac{2}{3 + i}$

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

$\frac{15 + 5 i}{9 + 3 i - 3 i - (i^{1} i)} + \frac{2}{3 + i}$

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

$\frac{15 + 5 i}{9 + 3 i - 3 i - (i^{1} i^{1})} + \frac{2}{3 + i}$

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

$\frac{15 + 5 i}{9 + 3 i - 3 i - i^{1 + 1}} + \frac{2}{3 + i}$

Add $1$ and $1$ .

$\frac{15 + 5 i}{9 + 3 i - 3 i - i^{2}} + \frac{2}{3 + i}$

Subtract $3 i$ from $3 i$ .

$\frac{15 + 5 i}{9 + 0 - i^{2}} + \frac{2}{3 + i}$

Add $9$ and $0$ .

$\frac{15 + 5 i}{9 - i^{2}} + \frac{2}{3 + i}$

Simplify each term.

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

$\frac{15 + 5 i}{9 - - 1} + \frac{2}{3 + i}$

Multiply $- 1$ by $- 1$ .

$\frac{15 + 5 i}{9 + 1} + \frac{2}{3 + i}$

Add $9$ and $1$ .

$\frac{15 + 5 i}{10} + \frac{2}{3 + i}$

Cancel the common factor of $15 + 5 i$ and $10$ .

Factor $5$ out of $15$ .

$\frac{5 \cdot 3 + 5 i}{10} + \frac{2}{3 + i}$

Factor $5$ out of $5 i$ .

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

Factor $5$ out of $5 \cdot 3 + 5 (i)$ .

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

Cancel the common factors.

Factor $5$ out of $10$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 + i}{2} + \frac{2}{3 + i}$

Split the fraction $\frac{3 + i}{2}$ into two fractions.

$\frac{3}{2} + \frac{i}{2} + \frac{2}{3 + i}$

Multiply the numerator and denominator of $\frac{2}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

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

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $3$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 + 2 (- i)}{(3 + 1 i) (3 - i)}$

Multiply $- 1$ by $2$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{(3 + 1 i) (3 - i)}$

Simplify the denominator.

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{3 (3 - i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Multiply $- 1$ by $3$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Multiply $3$ by $1$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - 3 i + 3 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - 3 i + 3 i - i i}$

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

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - 3 i + 3 i - (i^{1} i)}$

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

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - 3 i + 3 i - (i^{1} i^{1})}$

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

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - 3 i + 3 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - 3 i + 3 i - i^{2}}$

Add $- 3 i$ and $3 i$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 + 0 - i^{2}}$

Add $9$ and $0$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - i^{2}}$

Simplify each term.

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

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{9 + 1}$

Add $9$ and $1$ .

$\frac{3}{2} + \frac{i}{2} + \frac{6 - 2 i}{10}$

Cancel the common factor of $6 - 2 i$ and $10$ .

Factor $2$ out of $6$ .

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

Factor $2$ out of $- 2 i$ .

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

Factor $2$ out of $2 (3) + 2 (- i)$ .

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

Cancel the common factors.

Factor $2$ out of $10$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Split the fraction $\frac{3 - i}{5}$ into two fractions.

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

Move the negative in front of the fraction.

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

Step 2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 3

To write $\frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{2}$ by $\frac{5}{5}$ .

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

Multiply $2$ by $5$ .

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

Multiply $\frac{3}{5}$ by $\frac{2}{2}$ .

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

Multiply $5$ by $2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Multiply $3$ by $5$ .

$\frac{15 + 3 \cdot 2}{10} + \frac{i}{2} - \frac{i}{5}$

Multiply $3$ by $2$ .

$\frac{15 + 6}{10} + \frac{i}{2} - \frac{i}{5}$

Add $15$ and $6$ .

$\frac{21}{10} + \frac{i}{2} - \frac{i}{5}$

Step 7

To write $\frac{i}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{21}{10} + \frac{i}{2} \cdot \frac{5}{5} - \frac{i}{5}$

Step 8

To write $- \frac{i}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{21}{10} + \frac{i}{2} \cdot \frac{5}{5} - \frac{i}{5} \cdot \frac{2}{2}$

Step 9

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{i}{2}$ by $\frac{5}{5}$ .

$\frac{21}{10} + \frac{i \cdot 5}{2 \cdot 5} - \frac{i}{5} \cdot \frac{2}{2}$

Multiply $2$ by $5$ .

$\frac{21}{10} + \frac{i \cdot 5}{10} - \frac{i}{5} \cdot \frac{2}{2}$

Multiply $\frac{i}{5}$ by $\frac{2}{2}$ .

$\frac{21}{10} + \frac{i \cdot 5}{10} - \frac{i \cdot 2}{5 \cdot 2}$

Multiply $5$ by $2$ .

$\frac{21}{10} + \frac{i \cdot 5}{10} - \frac{i \cdot 2}{10}$

Step 10

Combine the numerators over the common denominator.

$\frac{21}{10} + \frac{3 i}{10}$

Simplify 5/(3-i)+2/(3+i)

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