Simplify the Radical Expression (5+6i)/(4+6i)

$\frac{5 + 6 i}{4 + 6 i}$

Step 1

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

$\frac{5 + 6 i}{4 + 6 i} \cdot \frac{4 - 6 i}{4 - 6 i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $4$ .

$\frac{20 + 5 (- 6 i) + 6 i \cdot 4 + 6 i (- 6 i)}{(4 + 6 i) (4 - 6 i)}$

Multiply $- 6$ by $5$ .

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

Multiply $4$ by $6$ .

$\frac{20 - 30 i + 24 i + 6 i (- 6 i)}{(4 + 6 i) (4 - 6 i)}$

Multiply $6 i (- 6 i)$ .

Multiply $- 6$ by $6$ .

$\frac{20 - 30 i + 24 i - 36 i i}{(4 + 6 i) (4 - 6 i)}$

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

$\frac{20 - 30 i + 24 i - 36 (i^{1} i)}{(4 + 6 i) (4 - 6 i)}$

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

$\frac{20 - 30 i + 24 i - 36 (i^{1} i^{1})}{(4 + 6 i) (4 - 6 i)}$

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

$\frac{20 - 30 i + 24 i - 36 i^{1 + 1}}{(4 + 6 i) (4 - 6 i)}$

Add $1$ and $1$ .

$\frac{20 - 30 i + 24 i - 36 i^{2}}{(4 + 6 i) (4 - 6 i)}$

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

$\frac{20 - 30 i + 24 i - 36 \cdot - 1}{(4 + 6 i) (4 - 6 i)}$

Multiply $- 36$ by $- 1$ .

$\frac{20 - 30 i + 24 i + 36}{(4 + 6 i) (4 - 6 i)}$

Add $20$ and $36$ .

$\frac{56 - 30 i + 24 i}{(4 + 6 i) (4 - 6 i)}$

Add $- 30 i$ and $24 i$ .

$\frac{56 - 6 i}{(4 + 6 i) (4 - 6 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{56 - 6 i}{4 (4 - 6 i) + 6 i (4 - 6 i)}$

Apply the distributive property.

$\frac{56 - 6 i}{4 \cdot 4 + 4 (- 6 i) + 6 i (4 - 6 i)}$

Apply the distributive property.

$\frac{56 - 6 i}{4 \cdot 4 + 4 (- 6 i) + 6 i \cdot 4 + 6 i (- 6 i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{56 - 6 i}{16 + 4 (- 6 i) + 6 i \cdot 4 + 6 i (- 6 i)}$

Multiply $- 6$ by $4$ .

$\frac{56 - 6 i}{16 - 24 i + 6 i \cdot 4 + 6 i (- 6 i)}$

Multiply $4$ by $6$ .

$\frac{56 - 6 i}{16 - 24 i + 24 i + 6 i (- 6 i)}$

Multiply $- 6$ by $6$ .

$\frac{56 - 6 i}{16 - 24 i + 24 i - 36 i i}$

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

$\frac{56 - 6 i}{16 - 24 i + 24 i - 36 (i^{1} i)}$

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

$\frac{56 - 6 i}{16 - 24 i + 24 i - 36 (i^{1} i^{1})}$

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

$\frac{56 - 6 i}{16 - 24 i + 24 i - 36 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{56 - 6 i}{16 - 24 i + 24 i - 36 i^{2}}$

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

$\frac{56 - 6 i}{16 + 0 - 36 i^{2}}$

Add $16$ and $0$ .

$\frac{56 - 6 i}{16 - 36 i^{2}}$

Simplify each term.

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

$\frac{56 - 6 i}{16 - 36 \cdot - 1}$

Multiply $- 36$ by $- 1$ .

$\frac{56 - 6 i}{16 + 36}$

Add $16$ and $36$ .

$\frac{56 - 6 i}{52}$

Step 3

Cancel the common factor of $56 - 6 i$ and $52$ .

Factor $2$ out of $56$ .

$\frac{2 (28) - 6 i}{52}$

Factor $2$ out of $- 6 i$ .

$\frac{2 (28) + 2 (- 3 i)}{52}$

Factor $2$ out of $2 (28) + 2 (- 3 i)$ .

$\frac{2 (28 - 3 i)}{52}$

Cancel the common factors.

Factor $2$ out of $52$ .

$\frac{2 (28 - 3 i)}{2 \cdot 26}$

Cancel the common factor.

$\frac{2 (28 - 3 i)}{2 \cdot 26}$

Rewrite the expression.

$\frac{28 - 3 i}{26}$

Step 4

Split the fraction $\frac{28 - 3 i}{26}$ into two fractions.

$\frac{28}{26} + \frac{- 3 i}{26}$

Step 5

Simplify each term.

Cancel the common factor of $28$ and $26$ .

Factor $2$ out of $28$ .

$\frac{2 (14)}{26} + \frac{- 3 i}{26}$

Cancel the common factors.

Factor $2$ out of $26$ .

$\frac{2 \cdot 14}{2 \cdot 13} + \frac{- 3 i}{26}$

Cancel the common factor.

$\frac{2 \cdot 14}{2 \cdot 13} + \frac{- 3 i}{26}$

Rewrite the expression.

$\frac{14}{13} + \frac{- 3 i}{26}$

Move the negative in front of the fraction.

$\frac{14}{13} - \frac{3 i}{26}$

Simplify the Radical Expression (5+6i)/(4+6i)

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