Precalculus Examples

$\frac{4 i}{1 - i} - \frac{8 + i}{2 + 3 i}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\frac{4 i}{1 - i} \cdot \frac{1 + i}{1 + i} - \frac{8 + i}{2 + 3 i}$

Step 1.2

Multiply.

Tap for more steps...

Step 1.2.1

Combine.

$\frac{4 i (1 + i)}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2

Simplify the numerator.

Tap for more steps...

Step 1.2.2.1

Apply the distributive property.

$\frac{4 i \cdot 1 + 4 i i}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.2

Multiply $4$ by $1$ .

$\frac{4 i + 4 i i}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.3

Multiply $4 i i$ .

Tap for more steps...

Step 1.2.2.3.1

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

$\frac{4 i + 4 (i^{1} i)}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.3.2

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

$\frac{4 i + 4 (i^{1} i^{1})}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.3.3

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

$\frac{4 i + 4 i^{1 + 1}}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.3.4

Add $1$ and $1$ .

$\frac{4 i + 4 i^{2}}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.4

Simplify each term.

Tap for more steps...

Step 1.2.2.4.1

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

$\frac{4 i + 4 \cdot - 1}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.4.2

Multiply $4$ by $- 1$ .

$\frac{4 i - 4}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.2.5

Reorder $4 i$ and $- 4$ .

$\frac{- 4 + 4 i}{(1 - i) (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3

Simplify the denominator.

Tap for more steps...

Step 1.2.3.1

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

Tap for more steps...

Step 1.2.3.1.1

Apply the distributive property.

$\frac{- 4 + 4 i}{1 (1 + i) - i (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.1.2

Apply the distributive property.

$\frac{- 4 + 4 i}{1 \cdot 1 + 1 i - i (1 + i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.1.3

Apply the distributive property.

$\frac{- 4 + 4 i}{1 \cdot 1 + 1 i - i \cdot 1 - i i} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2

Simplify.

Tap for more steps...

Step 1.2.3.2.1

Multiply $1$ by $1$ .

$\frac{- 4 + 4 i}{1 + 1 i - i \cdot 1 - i i} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2.2

Multiply $- 1$ by $1$ .

$\frac{- 4 + 4 i}{1 + 1 i - i - i i} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2.3

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

$\frac{- 4 + 4 i}{1 + 1 i - i - (i^{1} i)} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2.4

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

$\frac{- 4 + 4 i}{1 + 1 i - i - (i^{1} i^{1})} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2.5

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

$\frac{- 4 + 4 i}{1 + 1 i - i - i^{1 + 1}} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2.6

Add $1$ and $1$ .

$\frac{- 4 + 4 i}{1 + 1 i - i - i^{2}} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2.7

Subtract $i$ from $1 i$ .

$\frac{- 4 + 4 i}{1 + 0 - i^{2}} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.2.8

Add $1$ and $0$ .

$\frac{- 4 + 4 i}{1 - i^{2}} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.3

Simplify each term.

Tap for more steps...

Step 1.2.3.3.1

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

$\frac{- 4 + 4 i}{1 - - 1} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.3.2

Multiply $- 1$ by $- 1$ .

$\frac{- 4 + 4 i}{1 + 1} - \frac{8 + i}{2 + 3 i}$

Step 1.2.3.4

Add $1$ and $1$ .

$\frac{- 4 + 4 i}{2} - \frac{8 + i}{2 + 3 i}$

Step 1.3

Cancel the common factor of $- 4 + 4 i$ and $2$ .

Tap for more steps...

Step 1.3.1

Factor $2$ out of $- 4$ .

$\frac{2 \cdot - 2 + 4 i}{2} - \frac{8 + i}{2 + 3 i}$

Step 1.3.2

Factor $2$ out of $4 i$ .

$\frac{2 \cdot - 2 + 2 (2 i)}{2} - \frac{8 + i}{2 + 3 i}$

Step 1.3.3

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

$\frac{2 (- 2 + 2 i)}{2} - \frac{8 + i}{2 + 3 i}$

Step 1.3.4

Cancel the common factors.

Tap for more steps...

Step 1.3.4.1

Factor $2$ out of $2$ .

$\frac{2 (- 2 + 2 i)}{2 (1)} - \frac{8 + i}{2 + 3 i}$

Step 1.3.4.2

Cancel the common factor.

$\frac{2 (- 2 + 2 i)}{2 \cdot 1} - \frac{8 + i}{2 + 3 i}$

Step 1.3.4.3

Rewrite the expression.

$\frac{- 2 + 2 i}{1} - \frac{8 + i}{2 + 3 i}$

Step 1.3.4.4

Divide $- 2 + 2 i$ by $1$ .

$- 2 + 2 i - \frac{8 + i}{2 + 3 i}$

Step 1.4

Multiply the numerator and denominator of $\frac{8 + i}{2 + 3 i}$ by the conjugate of $2 + 3 i$ to make the denominator real.

$- 2 + 2 i - (\frac{8 + i}{2 + 3 i} \cdot \frac{2 - 3 i}{2 - 3 i})$

Step 1.5

Multiply.

Tap for more steps...

Step 1.5.1

Combine.

$- 2 + 2 i - \frac{(8 + i) (2 - 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2

Simplify the numerator.

Tap for more steps...

Step 1.5.2.1

Expand $(8 + i) (2 - 3 i)$ using the FOIL Method.

Tap for more steps...

Step 1.5.2.1.1

Apply the distributive property.

$- 2 + 2 i - \frac{8 (2 - 3 i) + i (2 - 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.1.2

Apply the distributive property.

$- 2 + 2 i - \frac{8 \cdot 2 + 8 (- 3 i) + i (2 - 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.1.3

Apply the distributive property.

$- 2 + 2 i - \frac{8 \cdot 2 + 8 (- 3 i) + i \cdot 2 + i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2

Simplify and combine like terms.

Tap for more steps...

Step 1.5.2.2.1

Simplify each term.

Tap for more steps...

Step 1.5.2.2.1.1

Multiply $8$ by $2$ .

$- 2 + 2 i - \frac{16 + 8 (- 3 i) + i \cdot 2 + i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.2

Multiply $- 3$ by $8$ .

$- 2 + 2 i - \frac{16 - 24 i + i \cdot 2 + i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.3

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

$- 2 + 2 i - \frac{16 - 24 i + 2 \cdot i + i (- 3 i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.4

Multiply $i (- 3 i)$ .

Tap for more steps...

Step 1.5.2.2.1.4.1

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

$- 2 + 2 i - \frac{16 - 24 i + 2 i - 3 (i^{1} i)}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.4.2

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

$- 2 + 2 i - \frac{16 - 24 i + 2 i - 3 (i^{1} i^{1})}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.4.3

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

$- 2 + 2 i - \frac{16 - 24 i + 2 i - 3 i^{1 + 1}}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.4.4

Add $1$ and $1$ .

$- 2 + 2 i - \frac{16 - 24 i + 2 i - 3 i^{2}}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.5

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

$- 2 + 2 i - \frac{16 - 24 i + 2 i - 3 \cdot - 1}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.1.6

Multiply $- 3$ by $- 1$ .

$- 2 + 2 i - \frac{16 - 24 i + 2 i + 3}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.2

Add $16$ and $3$ .

$- 2 + 2 i - \frac{19 - 24 i + 2 i}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.2.2.3

Add $- 24 i$ and $2 i$ .

$- 2 + 2 i - \frac{19 - 22 i}{(2 + 3 i) (2 - 3 i)}$

Step 1.5.3

Simplify the denominator.

Tap for more steps...

Step 1.5.3.1

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

Tap for more steps...

Step 1.5.3.1.1

Apply the distributive property.

$- 2 + 2 i - \frac{19 - 22 i}{2 (2 - 3 i) + 3 i (2 - 3 i)}$

Step 1.5.3.1.2

Apply the distributive property.

$- 2 + 2 i - \frac{19 - 22 i}{2 \cdot 2 + 2 (- 3 i) + 3 i (2 - 3 i)}$

Step 1.5.3.1.3

Apply the distributive property.

$- 2 + 2 i - \frac{19 - 22 i}{2 \cdot 2 + 2 (- 3 i) + 3 i \cdot 2 + 3 i (- 3 i)}$

Step 1.5.3.2

Simplify.

Tap for more steps...

Step 1.5.3.2.1

Multiply $2$ by $2$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 + 2 (- 3 i) + 3 i \cdot 2 + 3 i (- 3 i)}$

Step 1.5.3.2.2

Multiply $- 3$ by $2$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 - 6 i + 3 i \cdot 2 + 3 i (- 3 i)}$

Step 1.5.3.2.3

Multiply $2$ by $3$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 - 6 i + 6 i + 3 i (- 3 i)}$

Step 1.5.3.2.4

Multiply $- 3$ by $3$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 - 6 i + 6 i - 9 i i}$

Step 1.5.3.2.5

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

$- 2 + 2 i - \frac{19 - 22 i}{4 - 6 i + 6 i - 9 (i^{1} i)}$

Step 1.5.3.2.6

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

$- 2 + 2 i - \frac{19 - 22 i}{4 - 6 i + 6 i - 9 (i^{1} i^{1})}$

Step 1.5.3.2.7

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

$- 2 + 2 i - \frac{19 - 22 i}{4 - 6 i + 6 i - 9 i^{1 + 1}}$

Step 1.5.3.2.8

Add $1$ and $1$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 - 6 i + 6 i - 9 i^{2}}$

Step 1.5.3.2.9

Add $- 6 i$ and $6 i$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 + 0 - 9 i^{2}}$

Step 1.5.3.2.10

Add $4$ and $0$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 - 9 i^{2}}$

Step 1.5.3.3

Simplify each term.

Tap for more steps...

Step 1.5.3.3.1

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

$- 2 + 2 i - \frac{19 - 22 i}{4 - 9 \cdot - 1}$

Step 1.5.3.3.2

Multiply $- 9$ by $- 1$ .

$- 2 + 2 i - \frac{19 - 22 i}{4 + 9}$

Step 1.5.3.4

Add $4$ and $9$ .

$- 2 + 2 i - \frac{19 - 22 i}{13}$

Step 1.6

Split the fraction $\frac{19 - 22 i}{13}$ into two fractions.

$- 2 + 2 i - (\frac{19}{13} + \frac{- 22 i}{13})$

Step 1.7

Move the negative in front of the fraction.

$- 2 + 2 i - (\frac{19}{13} - \frac{22 i}{13})$

Step 1.8

Apply the distributive property.

$- 2 + 2 i - \frac{19}{13} - - \frac{22 i}{13}$

Step 1.9

Multiply $- - \frac{22 i}{13}$ .

Tap for more steps...

Step 1.9.1

Multiply $- 1$ by $- 1$ .

$- 2 + 2 i - \frac{19}{13} + 1 \frac{22 i}{13}$

Step 1.9.2

Multiply $\frac{22 i}{13}$ by $1$ .

$- 2 + 2 i - \frac{19}{13} + \frac{22 i}{13}$

Step 2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$- 2 \cdot \frac{13}{13} - \frac{19}{13} + 2 i + \frac{22 i}{13}$

Step 3

Combine $- 2$ and $\frac{13}{13}$ .

$\frac{- 2 \cdot 13}{13} - \frac{19}{13} + 2 i + \frac{22 i}{13}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 13 - 19}{13} + 2 i + \frac{22 i}{13}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Multiply $- 2$ by $13$ .

$\frac{- 26 - 19}{13} + 2 i + \frac{22 i}{13}$

Step 5.2

Subtract $19$ from $- 26$ .

$\frac{- 45}{13} + 2 i + \frac{22 i}{13}$

Step 6

Move the negative in front of the fraction.

$- \frac{45}{13} + 2 i + \frac{22 i}{13}$

Step 7

To write $2 i$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$- \frac{45}{13} + 2 i \cdot \frac{13}{13} + \frac{22 i}{13}$

Step 8

Combine $2 i$ and $\frac{13}{13}$ .

$- \frac{45}{13} + \frac{2 i \cdot 13}{13} + \frac{22 i}{13}$

Step 9

Combine the numerators over the common denominator.

$- \frac{45}{13} + \frac{48 i}{13}$