Simplify (1+i)/(4-3i)

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

Step 1

Multiply the numerator and denominator of $\frac{1 + i}{4 - 3 i}$ by the conjugate of $4 - 3 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(1 + i) (4 + 3 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $1$ .

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

Multiply $3 i$ by $1$ .

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

Move $4$ to the left of $i$ .

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

Multiply $i (3 i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $3$ by $- 1$ .

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

Subtract $3$ from $4$ .

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

Add $3 i$ and $4 i$ .

$\frac{1 + 7 i}{(4 - 3 i) (4 + 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{1 + 7 i}{4 (4 + 3 i) - 3 i (4 + 3 i)}$

Apply the distributive property.

$\frac{1 + 7 i}{4 \cdot 4 + 4 (3 i) - 3 i (4 + 3 i)}$

Apply the distributive property.

$\frac{1 + 7 i}{4 \cdot 4 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{1 + 7 i}{16 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Multiply $3$ by $4$ .

$\frac{1 + 7 i}{16 + 12 i - 3 i \cdot 4 - 3 i (3 i)}$

Multiply $4$ by $- 3$ .

$\frac{1 + 7 i}{16 + 12 i - 12 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{1 + 7 i}{16 + 12 i - 12 i - 9 i i}$

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

$\frac{1 + 7 i}{16 + 12 i - 12 i - 9 (i^{1} i)}$

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

$\frac{1 + 7 i}{16 + 12 i - 12 i - 9 (i^{1} i^{1})}$

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

$\frac{1 + 7 i}{16 + 12 i - 12 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{1 + 7 i}{16 + 12 i - 12 i - 9 i^{2}}$

Subtract $12 i$ from $12 i$ .

$\frac{1 + 7 i}{16 + 0 - 9 i^{2}}$

Add $16$ and $0$ .

$\frac{1 + 7 i}{16 - 9 i^{2}}$

Simplify each term.

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

$\frac{1 + 7 i}{16 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{1 + 7 i}{16 + 9}$

Add $16$ and $9$ .

$\frac{1 + 7 i}{25}$

Step 3

Split the fraction $\frac{1 + 7 i}{25}$ into two fractions.

$\frac{1}{25} + \frac{7 i}{25}$

Simplify (1+i)/(4-3i)

Related Questions

Support LCMGCF.com by disabling your adblocker.