Simplify 37/(6+i)

$\frac{37}{6 + i}$

Step 1

Multiply the numerator and denominator of $\frac{37}{6 + 1 i}$ by the conjugate of $6 + 1 i$ to make the denominator real.

$\frac{37}{6 + 1 i} \cdot \frac{6 - i}{6 - i}$

Step 2

Multiply.

Combine.

$\frac{37 (6 - i)}{(6 + 1 i) (6 - i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{37 \cdot 6 + 37 (- i)}{(6 + 1 i) (6 - i)}$

Multiply $37$ by $6$ .

$\frac{222 + 37 (- i)}{(6 + 1 i) (6 - i)}$

Multiply $- 1$ by $37$ .

$\frac{222 - 37 i}{(6 + 1 i) (6 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{222 - 37 i}{6 (6 - i) + 1 i (6 - i)}$

Apply the distributive property.

$\frac{222 - 37 i}{6 \cdot 6 + 6 (- i) + 1 i (6 - i)}$

Apply the distributive property.

$\frac{222 - 37 i}{6 \cdot 6 + 6 (- i) + 1 i \cdot 6 + 1 i (- i)}$

Simplify.

Multiply $6$ by $6$ .

$\frac{222 - 37 i}{36 + 6 (- i) + 1 i \cdot 6 + 1 i (- i)}$

Multiply $- 1$ by $6$ .

$\frac{222 - 37 i}{36 - 6 i + 1 i \cdot 6 + 1 i (- i)}$

Multiply $6$ by $1$ .

$\frac{222 - 37 i}{36 - 6 i + 6 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{222 - 37 i}{36 - 6 i + 6 i - i i}$

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

$\frac{222 - 37 i}{36 - 6 i + 6 i - (i^{1} i)}$

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

$\frac{222 - 37 i}{36 - 6 i + 6 i - (i^{1} i^{1})}$

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

$\frac{222 - 37 i}{36 - 6 i + 6 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{222 - 37 i}{36 - 6 i + 6 i - i^{2}}$

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

$\frac{222 - 37 i}{36 + 0 - i^{2}}$

Add $36$ and $0$ .

$\frac{222 - 37 i}{36 - i^{2}}$

Simplify each term.

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

$\frac{222 - 37 i}{36 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{222 - 37 i}{36 + 1}$

Add $36$ and $1$ .

$\frac{222 - 37 i}{37}$

Step 3

Cancel the common factor of $222 - 37 i$ and $37$ .

Factor $37$ out of $222$ .

$\frac{37 \cdot 6 - 37 i}{37}$

Factor $37$ out of $- 37 i$ .

$\frac{37 \cdot 6 + 37 (- i)}{37}$

Factor $37$ out of $37 (6) + 37 (- i)$ .

$\frac{37 (6 - i)}{37}$

Cancel the common factors.

Factor $37$ out of $37$ .

$\frac{37 (6 - i)}{37 (1)}$

Cancel the common factor.

$\frac{37 (6 - i)}{37 \cdot 1}$

Rewrite the expression.

$\frac{6 - i}{1}$

Divide $6 - i$ by $1$ .

$6 - i$

Simplify 37/(6+i)

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