Algebra Examples

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

Step 1

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

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

Step 2

Multiply.

Tap for more steps...

Step 2.1

Combine.

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

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Expand $(3 + i) (2 + i)$ using the FOIL Method.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Multiply $3$ by $2$ .

$\frac{6 + 3 i + i \cdot 2 + i i}{(2 - i) (2 + i)} \cdot \frac{4 + i}{2 + i}$

Step 2.2.2.1.2

Move $2$ to the left of $i$ .

$\frac{6 + 3 i + 2 \cdot i + i i}{(2 - i) (2 + i)} \cdot \frac{4 + i}{2 + i}$

Step 2.2.2.1.3

Multiply $i i$ .

Tap for more steps...

Step 2.2.2.1.3.1

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

$\frac{6 + 3 i + 2 i + i^{1} i}{(2 - i) (2 + i)} \cdot \frac{4 + i}{2 + i}$

Step 2.2.2.1.3.2

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

$\frac{6 + 3 i + 2 i + i^{1} i^{1}}{(2 - i) (2 + i)} \cdot \frac{4 + i}{2 + i}$

Step 2.2.2.1.3.3

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

$\frac{6 + 3 i + 2 i + i^{1 + 1}}{(2 - i) (2 + i)} \cdot \frac{4 + i}{2 + i}$

Step 2.2.2.1.3.4

Add $1$ and $1$ .

$\frac{6 + 3 i + 2 i + i^{2}}{(2 - i) (2 + i)} \cdot \frac{4 + i}{2 + i}$

Step 2.2.2.1.4

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

$\frac{6 + 3 i + 2 i - 1}{(2 - i) (2 + i)} \cdot \frac{4 + i}{2 + i}$

Step 2.2.2.2

Subtract $1$ from $6$ .

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

Step 2.2.2.3

Add $3 i$ and $2 i$ .

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

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

Tap for more steps...

Step 2.3.2.1

Multiply $2$ by $2$ .

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

Step 2.3.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2.3

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

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

Step 2.3.2.4

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

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

Step 2.3.2.5

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

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

Step 2.3.2.6

Add $1$ and $1$ .

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

Step 2.3.2.7

Subtract $2 i$ from $2 i$ .

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

Step 2.3.2.8

Add $4$ and $0$ .

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

Step 2.3.3

Simplify each term.

Tap for more steps...

Step 2.3.3.1

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

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

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4

Add $4$ and $1$ .

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

Step 3

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

Tap for more steps...

Step 3.1

Factor $5$ out of $5$ .

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

Step 3.2

Factor $5$ out of $5 i$ .

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

Step 3.3

Factor $5$ out of $5 (1) + 5 (i)$ .

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

Step 3.4

Cancel the common factors.

Tap for more steps...

Step 3.4.1

Factor $5$ out of $5$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.4.4

Divide $1 + i$ by $1$ .

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

Step 4

Multiply the numerator and denominator of $\frac{4 + i}{2 + 1 i}$ by the conjugate of $2 + 1 i$ to make the denominator real.

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

Step 5

Multiply.

Tap for more steps...

Step 5.1

Combine.

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

Step 5.2

Simplify the numerator.

Tap for more steps...

Step 5.2.1

Expand $(4 + i) (2 - i)$ using the FOIL Method.

Tap for more steps...

Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Apply the distributive property.

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

Step 5.2.2

Simplify and combine like terms.

Tap for more steps...

Step 5.2.2.1

Simplify each term.

Tap for more steps...

Step 5.2.2.1.1

Multiply $4$ by $2$ .

$(1 + i) \cdot \frac{8 + 4 (- i) + i \cdot 2 + i (- i)}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.2

Multiply $- 1$ by $4$ .

$(1 + i) \cdot \frac{8 - 4 i + i \cdot 2 + i (- i)}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.3

Move $2$ to the left of $i$ .

$(1 + i) \cdot \frac{8 - 4 i + 2 \cdot i + i (- i)}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.4

Multiply $i (- i)$ .

Tap for more steps...

Step 5.2.2.1.4.1

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

$(1 + i) \cdot \frac{8 - 4 i + 2 i - (i^{1} i)}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.4.2

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

$(1 + i) \cdot \frac{8 - 4 i + 2 i - (i^{1} i^{1})}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.4.3

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

$(1 + i) \cdot \frac{8 - 4 i + 2 i - i^{1 + 1}}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.4.4

Add $1$ and $1$ .

$(1 + i) \cdot \frac{8 - 4 i + 2 i - i^{2}}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.5

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

$(1 + i) \cdot \frac{8 - 4 i + 2 i - - 1}{(2 + 1 i) (2 - i)}$

Step 5.2.2.1.6

Multiply $- 1$ by $- 1$ .

$(1 + i) \cdot \frac{8 - 4 i + 2 i + 1}{(2 + 1 i) (2 - i)}$

Step 5.2.2.2

Add $8$ and $1$ .

$(1 + i) \cdot \frac{9 - 4 i + 2 i}{(2 + 1 i) (2 - i)}$

Step 5.2.2.3

Add $- 4 i$ and $2 i$ .

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

Step 5.3

Simplify the denominator.

Tap for more steps...

Step 5.3.1

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

Tap for more steps...

Step 5.3.1.1

Apply the distributive property.

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

Step 5.3.1.2

Apply the distributive property.

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

Step 5.3.1.3

Apply the distributive property.

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

Step 5.3.2

Simplify.

Tap for more steps...

Step 5.3.2.1

Multiply $2$ by $2$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Step 5.3.2.2

Multiply $- 1$ by $2$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)}$

Step 5.3.2.3

Multiply $2$ by $1$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 - 2 i + 2 i + 1 i (- i)}$

Step 5.3.2.4

Multiply $- 1$ by $1$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 - 2 i + 2 i - i i}$

Step 5.3.2.5

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

$(1 + i) \cdot \frac{9 - 2 i}{4 - 2 i + 2 i - (i^{1} i)}$

Step 5.3.2.6

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

$(1 + i) \cdot \frac{9 - 2 i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

Step 5.3.2.7

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

$(1 + i) \cdot \frac{9 - 2 i}{4 - 2 i + 2 i - i^{1 + 1}}$

Step 5.3.2.8

Add $1$ and $1$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 - 2 i + 2 i - i^{2}}$

Step 5.3.2.9

Add $- 2 i$ and $2 i$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 + 0 - i^{2}}$

Step 5.3.2.10

Add $4$ and $0$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 - i^{2}}$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

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

$(1 + i) \cdot \frac{9 - 2 i}{4 - - 1}$

Step 5.3.3.2

Multiply $- 1$ by $- 1$ .

$(1 + i) \cdot \frac{9 - 2 i}{4 + 1}$

Step 5.3.4

Add $4$ and $1$ .

$(1 + i) \cdot \frac{9 - 2 i}{5}$

Step 6

Split the fraction $\frac{9 - 2 i}{5}$ into two fractions.

$(1 + i) \cdot (\frac{9}{5} + \frac{- 2 i}{5})$

Step 7

Move the negative in front of the fraction.

$(1 + i) \cdot (\frac{9}{5} - \frac{2 i}{5})$

Step 8

Expand $(1 + i) (\frac{9}{5} - \frac{2 i}{5})$ using the FOIL Method.

Tap for more steps...

Step 8.1

Apply the distributive property.

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Apply the distributive property.

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

Step 9

Simplify and combine like terms.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Multiply $\frac{9}{5}$ by $1$ .

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

Step 9.1.2

Multiply $- \frac{2 i}{5}$ by $1$ .

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

Step 9.1.3

Combine $i$ and $\frac{9}{5}$ .

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

Step 9.1.4

Move $9$ to the left of $i$ .

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

Step 9.1.5

Multiply $i (- \frac{2 i}{5})$ .

Tap for more steps...

Step 9.1.5.1

Combine $i$ and $\frac{2 i}{5}$ .

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

Step 9.1.5.2

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

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

Step 9.1.5.3

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

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

Step 9.1.5.4

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

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

Step 9.1.5.5

Add $1$ and $1$ .

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

Step 9.1.6

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

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

Step 9.1.7

Multiply $2$ by $- 1$ .

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

Step 9.1.8

Move the negative in front of the fraction.

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

Step 9.1.9

Multiply $- - \frac{2}{5}$ .

Tap for more steps...

Step 9.1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 9.1.9.2

Multiply $\frac{2}{5}$ by $1$ .

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

Step 9.2

Combine the numerators over the common denominator.

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

Step 9.3

Add $9$ and $2$ .

$\frac{11}{5} - \frac{2 i}{5} + \frac{9 i}{5}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{11}{5} + \frac{7 i}{5}$