Simplify (5+2i)/(4i)

$\frac{5 + 2 i}{4 i}$

Step 1

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

$\frac{5 + 2 i}{4 i} \cdot \frac{i}{i}$

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Apply the distributive property.

$\frac{5 i + 2 i i}{4 i i}$

Multiply $2 i i$ .

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

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

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

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

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

$\frac{5 i + 2 i^{1 + 1}}{4 i i}$

Add $1$ and $1$ .

$\frac{5 i + 2 i^{2}}{4 i i}$

Simplify each term.

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

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

Multiply $2$ by $- 1$ .

$\frac{5 i - 2}{4 i i}$

Reorder $5 i$ and $- 2$ .

$\frac{- 2 + 5 i}{4 i i}$

Simplify the denominator.

Add parentheses.

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

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

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

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

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

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

$\frac{- 2 + 5 i}{4 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 2 + 5 i}{4 i^{2}}$

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

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

Step 3

Multiply $4$ by $- 1$ .

$\frac{- 2 + 5 i}{- 4}$

Step 4

Split the fraction $\frac{- 2 + 5 i}{- 4}$ into two fractions.

$\frac{- 2}{- 4} + \frac{5 i}{- 4}$

Step 5

Simplify each term.

Cancel the common factor of $- 2$ and $- 4$ .

Factor $- 2$ out of $- 2$ .

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

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

$\frac{1}{2} - \frac{5 i}{4}$

Simplify (5+2i)/(4i)

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