Write in Standard Form ((5+2i)(6-4i))/(2+i)

$\frac{(5 + 2 i) (6 - 4 i)}{2 + i}$

Step 1

Multiply the numerator and denominator of $\frac{(5 + 2 i) (6 - 4 i)}{2 + 1 i}$ by the conjugate of $2 + 1 i$ to make the denominator real.

$\frac{(5 + 2 i) (6 - 4 i)}{2 + 1 i} \cdot \frac{2 - i}{2 - i}$

Step 2

Multiply.

Combine.

$\frac{(5 + 2 i) (6 - 4 i) (2 - i)}{(2 + 1 i) (2 - i)}$

Simplify the numerator.

Expand $(5 + 2 i) (6 - 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{(5 (6 - 4 i) + 2 i (6 - 4 i)) (2 - i)}{(2 + 1 i) (2 - i)}$

Apply the distributive property.

$\frac{(5 \cdot 6 + 5 (- 4 i) + 2 i (6 - 4 i)) (2 - i)}{(2 + 1 i) (2 - i)}$

Apply the distributive property.

$\frac{(5 \cdot 6 + 5 (- 4 i) + 2 i \cdot 6 + 2 i (- 4 i)) (2 - i)}{(2 + 1 i) (2 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $6$ .

$\frac{(30 + 5 (- 4 i) + 2 i \cdot 6 + 2 i (- 4 i)) (2 - i)}{(2 + 1 i) (2 - i)}$

Multiply $- 4$ by $5$ .

$\frac{(30 - 20 i + 2 i \cdot 6 + 2 i (- 4 i)) (2 - i)}{(2 + 1 i) (2 - i)}$

Multiply $6$ by $2$ .

$\frac{(30 - 20 i + 12 i + 2 i (- 4 i)) (2 - i)}{(2 + 1 i) (2 - i)}$

Multiply $2 i (- 4 i)$ .

Multiply $- 4$ by $2$ .

$\frac{(30 - 20 i + 12 i - 8 i i) (2 - i)}{(2 + 1 i) (2 - i)}$

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

$\frac{(30 - 20 i + 12 i - 8 (i^{1} i)) (2 - i)}{(2 + 1 i) (2 - i)}$

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

$\frac{(30 - 20 i + 12 i - 8 (i^{1} i^{1})) (2 - i)}{(2 + 1 i) (2 - i)}$

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

$\frac{(30 - 20 i + 12 i - 8 i^{1 + 1}) (2 - i)}{(2 + 1 i) (2 - i)}$

Add $1$ and $1$ .

$\frac{(30 - 20 i + 12 i - 8 i^{2}) (2 - i)}{(2 + 1 i) (2 - i)}$

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

$\frac{(30 - 20 i + 12 i - 8 \cdot - 1) (2 - i)}{(2 + 1 i) (2 - i)}$

Multiply $- 8$ by $- 1$ .

$\frac{(30 - 20 i + 12 i + 8) (2 - i)}{(2 + 1 i) (2 - i)}$

Add $30$ and $8$ .

$\frac{(38 - 20 i + 12 i) (2 - i)}{(2 + 1 i) (2 - i)}$

Add $- 20 i$ and $12 i$ .

$\frac{(38 - 8 i) (2 - i)}{(2 + 1 i) (2 - i)}$

Expand $(38 - 8 i) (2 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{38 (2 - i) - 8 i (2 - i)}{(2 + 1 i) (2 - i)}$

Apply the distributive property.

$\frac{38 \cdot 2 + 38 (- i) - 8 i (2 - i)}{(2 + 1 i) (2 - i)}$

Apply the distributive property.

$\frac{38 \cdot 2 + 38 (- i) - 8 i \cdot 2 - 8 i (- i)}{(2 + 1 i) (2 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $38$ by $2$ .

$\frac{76 + 38 (- i) - 8 i \cdot 2 - 8 i (- i)}{(2 + 1 i) (2 - i)}$

Multiply $- 1$ by $38$ .

$\frac{76 - 38 i - 8 i \cdot 2 - 8 i (- i)}{(2 + 1 i) (2 - i)}$

Multiply $2$ by $- 8$ .

$\frac{76 - 38 i - 16 i - 8 i (- i)}{(2 + 1 i) (2 - i)}$

Multiply $- 8 i (- i)$ .

Multiply $- 1$ by $- 8$ .

$\frac{76 - 38 i - 16 i + 8 i i}{(2 + 1 i) (2 - i)}$

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

$\frac{76 - 38 i - 16 i + 8 (i^{1} i)}{(2 + 1 i) (2 - i)}$

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

$\frac{76 - 38 i - 16 i + 8 (i^{1} i^{1})}{(2 + 1 i) (2 - i)}$

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

$\frac{76 - 38 i - 16 i + 8 i^{1 + 1}}{(2 + 1 i) (2 - i)}$

Add $1$ and $1$ .

$\frac{76 - 38 i - 16 i + 8 i^{2}}{(2 + 1 i) (2 - i)}$

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

$\frac{76 - 38 i - 16 i + 8 \cdot - 1}{(2 + 1 i) (2 - i)}$

Multiply $8$ by $- 1$ .

$\frac{76 - 38 i - 16 i - 8}{(2 + 1 i) (2 - i)}$

Subtract $8$ from $76$ .

$\frac{68 - 38 i - 16 i}{(2 + 1 i) (2 - i)}$

Subtract $16 i$ from $- 38 i$ .

$\frac{68 - 54 i}{(2 + 1 i) (2 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{68 - 54 i}{2 (2 - i) + 1 i (2 - i)}$

Apply the distributive property.

$\frac{68 - 54 i}{2 \cdot 2 + 2 (- i) + 1 i (2 - i)}$

Apply the distributive property.

$\frac{68 - 54 i}{2 \cdot 2 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{68 - 54 i}{4 + 2 (- i) + 1 i \cdot 2 + 1 i (- i)}$

Multiply $- 1$ by $2$ .

$\frac{68 - 54 i}{4 - 2 i + 1 i \cdot 2 + 1 i (- i)}$

Multiply $2$ by $1$ .

$\frac{68 - 54 i}{4 - 2 i + 2 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{68 - 54 i}{4 - 2 i + 2 i - i i}$

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

$\frac{68 - 54 i}{4 - 2 i + 2 i - (i^{1} i)}$

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

$\frac{68 - 54 i}{4 - 2 i + 2 i - (i^{1} i^{1})}$

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

$\frac{68 - 54 i}{4 - 2 i + 2 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{68 - 54 i}{4 - 2 i + 2 i - i^{2}}$

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

$\frac{68 - 54 i}{4 + 0 - i^{2}}$

Add $4$ and $0$ .

$\frac{68 - 54 i}{4 - i^{2}}$

Simplify each term.

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

$\frac{68 - 54 i}{4 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{68 - 54 i}{4 + 1}$

Add $4$ and $1$ .

$\frac{68 - 54 i}{5}$

Step 3

Split the fraction $\frac{68 - 54 i}{5}$ into two fractions.

$\frac{68}{5} + \frac{- 54 i}{5}$

Step 4

Move the negative in front of the fraction.

$\frac{68}{5} - \frac{54 i}{5}$

Write in Standard Form ((5+2i)(6-4i))/(2+i)

Related Questions

Support LCMGCF.com by disabling your adblocker.