Write in Standard Form (7+6i)/(6+i)

$\frac{7 + 6 i}{6 + i}$

Step 1

Multiply the numerator and denominator of $\frac{7 + 6 i}{6 + 1 i}$ by the conjugate of $6 + 1 i$ to make the denominator real.

$\frac{7 + 6 i}{6 + 1 i} \cdot \frac{6 - i}{6 - i}$

Step 2

Multiply.

Combine.

$\frac{(7 + 6 i) (6 - i)}{(6 + 1 i) (6 - i)}$

Simplify the numerator.

Expand $(7 + 6 i) (6 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{7 (6 - i) + 6 i (6 - i)}{(6 + 1 i) (6 - i)}$

Apply the distributive property.

$\frac{7 \cdot 6 + 7 (- i) + 6 i (6 - i)}{(6 + 1 i) (6 - i)}$

Apply the distributive property.

$\frac{7 \cdot 6 + 7 (- i) + 6 i \cdot 6 + 6 i (- i)}{(6 + 1 i) (6 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $6$ .

$\frac{42 + 7 (- i) + 6 i \cdot 6 + 6 i (- i)}{(6 + 1 i) (6 - i)}$

Multiply $- 1$ by $7$ .

$\frac{42 - 7 i + 6 i \cdot 6 + 6 i (- i)}{(6 + 1 i) (6 - i)}$

Multiply $6$ by $6$ .

$\frac{42 - 7 i + 36 i + 6 i (- i)}{(6 + 1 i) (6 - i)}$

Multiply $6 i (- i)$ .

Multiply $- 1$ by $6$ .

$\frac{42 - 7 i + 36 i - 6 i i}{(6 + 1 i) (6 - i)}$

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

$\frac{42 - 7 i + 36 i - 6 (i^{1} i)}{(6 + 1 i) (6 - i)}$

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

$\frac{42 - 7 i + 36 i - 6 (i^{1} i^{1})}{(6 + 1 i) (6 - i)}$

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

$\frac{42 - 7 i + 36 i - 6 i^{1 + 1}}{(6 + 1 i) (6 - i)}$

Add $1$ and $1$ .

$\frac{42 - 7 i + 36 i - 6 i^{2}}{(6 + 1 i) (6 - i)}$

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

$\frac{42 - 7 i + 36 i - 6 \cdot - 1}{(6 + 1 i) (6 - i)}$

Multiply $- 6$ by $- 1$ .

$\frac{42 - 7 i + 36 i + 6}{(6 + 1 i) (6 - i)}$

Add $42$ and $6$ .

$\frac{48 - 7 i + 36 i}{(6 + 1 i) (6 - i)}$

Add $- 7 i$ and $36 i$ .

$\frac{48 + 29 i}{(6 + 1 i) (6 - i)}$

Simplify the denominator.

Expand $(6 + 1 i) (6 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{48 + 29 i}{6 (6 - i) + 1 i (6 - i)}$

Apply the distributive property.

$\frac{48 + 29 i}{6 \cdot 6 + 6 (- i) + 1 i (6 - i)}$

Apply the distributive property.

$\frac{48 + 29 i}{6 \cdot 6 + 6 (- i) + 1 i \cdot 6 + 1 i (- i)}$

Simplify.

Multiply $6$ by $6$ .

$\frac{48 + 29 i}{36 + 6 (- i) + 1 i \cdot 6 + 1 i (- i)}$

Multiply $- 1$ by $6$ .

$\frac{48 + 29 i}{36 - 6 i + 1 i \cdot 6 + 1 i (- i)}$

Multiply $6$ by $1$ .

$\frac{48 + 29 i}{36 - 6 i + 6 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{48 + 29 i}{36 - 6 i + 6 i - i i}$

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

$\frac{48 + 29 i}{36 - 6 i + 6 i - (i^{1} i)}$

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

$\frac{48 + 29 i}{36 - 6 i + 6 i - (i^{1} i^{1})}$

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

$\frac{48 + 29 i}{36 - 6 i + 6 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{48 + 29 i}{36 - 6 i + 6 i - i^{2}}$

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

$\frac{48 + 29 i}{36 + 0 - i^{2}}$

Add $36$ and $0$ .

$\frac{48 + 29 i}{36 - i^{2}}$

Simplify each term.

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

$\frac{48 + 29 i}{36 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{48 + 29 i}{36 + 1}$

Add $36$ and $1$ .

$\frac{48 + 29 i}{37}$

Step 3

Split the fraction $\frac{48 + 29 i}{37}$ into two fractions.

$\frac{48}{37} + \frac{29 i}{37}$

Write in Standard Form (7+6i)/(6+i)

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