Write in Standard Form (1+2i)/( square root of 2+i)

$\frac{1 + 2 i}{\sqrt{2} + i}$

Multiply the numerator and denominator of $\frac{1 + 2 i}{1.41421356 + 1 i}$ by the conjugate of $1.41421356 + 1 i$ to make the denominator real.

$\frac{1 + 2 i}{1.41421356 + 1 i} \cdot \frac{1.41421356 - i}{1.41421356 - i}$

Multiply.

Combine.

$\frac{(1 + 2 i) (1.41421356 - i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Simplify the numerator.

Expand $(1 + 2 i) (1.41421356 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{1 (1.41421356 - i) + 2 i (1.41421356 - i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Apply the distributive property.

$\frac{1 \cdot 1.41421356 + 1 (- i) + 2 i (1.41421356 - i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Apply the distributive property.

$\frac{1 \cdot 1.41421356 + 1 (- i) + 2 i \cdot 1.41421356 + 2 i (- i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $1.41421356$ by $1$ .

$\frac{1.41421356 + 1 (- i) + 2 i \cdot 1.41421356 + 2 i (- i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Multiply $- i$ by $1$ .

$\frac{1.41421356 - i + 2 i \cdot 1.41421356 + 2 i (- i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Multiply $1.41421356$ by $2$ .

$\frac{1.41421356 - i + 2.82842712 i + 2 i (- i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

Multiply $2 i (- i)$ .

Multiply $- 1$ by $2$ .

$\frac{1.41421356 - i + 2.82842712 i - 2 i i}{(1.41421356 + 1 i) (1.41421356 - i)}$

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

$\frac{1.41421356 - i + 2.82842712 i - 2 (i^{1} i)}{(1.41421356 + 1 i) (1.41421356 - i)}$

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

$\frac{1.41421356 - i + 2.82842712 i - 2 (i^{1} i^{1})}{(1.41421356 + 1 i) (1.41421356 - i)}$

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

$\frac{1.41421356 - i + 2.82842712 i - 2 i^{1 + 1}}{(1.41421356 + 1 i) (1.41421356 - i)}$

Add $1$ and $1$ .

$\frac{1.41421356 - i + 2.82842712 i - 2 i^{2}}{(1.41421356 + 1 i) (1.41421356 - i)}$

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

$\frac{1.41421356 - i + 2.82842712 i - 2 \cdot - 1}{(1.41421356 + 1 i) (1.41421356 - i)}$

Multiply $- 2$ by $- 1$ .

$\frac{1.41421356 - i + 2.82842712 i + 2}{(1.41421356 + 1 i) (1.41421356 - i)}$

Add $1.41421356$ and $2$ .

$\frac{3.41421356 - i + 2.82842712 i}{(1.41421356 + 1 i) (1.41421356 - i)}$

Add $- i$ and $2.82842712 i$ .

$\frac{3.41421356 + 1.82842712 i}{(1.41421356 + 1 i) (1.41421356 - i)}$

Simplify the denominator.

Expand $(1.41421356 + 1 i) (1.41421356 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{3.41421356 + 1.82842712 i}{1.41421356 (1.41421356 - i) + 1 i (1.41421356 - i)}$

Apply the distributive property.

$\frac{3.41421356 + 1.82842712 i}{1.41421356 \cdot 1.41421356 + 1.41421356 (- i) + 1 i (1.41421356 - i)}$

Apply the distributive property.

$\frac{3.41421356 + 1.82842712 i}{1.41421356 \cdot 1.41421356 + 1.41421356 (- i) + 1 i \cdot 1.41421356 + 1 i (- i)}$

Simplify.

Multiply $1.41421356$ by $1.41421356$ .

$\frac{3.41421356 + 1.82842712 i}{2 + 1.41421356 (- i) + 1 i \cdot 1.41421356 + 1 i (- i)}$

Multiply $- 1$ by $1.41421356$ .

$\frac{3.41421356 + 1.82842712 i}{2 - 1.41421356 i + 1 i \cdot 1.41421356 + 1 i (- i)}$

Multiply $1.41421356$ by $1$ .

$\frac{3.41421356 + 1.82842712 i}{2 - 1.41421356 i + 1.41421356 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{3.41421356 + 1.82842712 i}{2 - 1.41421356 i + 1.41421356 i - i i}$

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

$\frac{3.41421356 + 1.82842712 i}{2 - 1.41421356 i + 1.41421356 i - (i^{1} i)}$

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

$\frac{3.41421356 + 1.82842712 i}{2 - 1.41421356 i + 1.41421356 i - (i^{1} i^{1})}$

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

$\frac{3.41421356 + 1.82842712 i}{2 - 1.41421356 i + 1.41421356 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3.41421356 + 1.82842712 i}{2 - 1.41421356 i + 1.41421356 i - i^{2}}$

Add $- 1.41421356 i$ and $1.41421356 i$ .

$\frac{3.41421356 + 1.82842712 i}{2 + 0 i - i^{2}}$

Simplify each term.

Multiply $0$ by $i$ .

$\frac{3.41421356 + 1.82842712 i}{2 + 0 - i^{2}}$

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

$\frac{3.41421356 + 1.82842712 i}{2 + 0 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{3.41421356 + 1.82842712 i}{2 + 0 + 1}$

Add $2$ and $0$ .

$\frac{3.41421356 + 1.82842712 i}{2 + 1}$

Add $2$ and $1$ .

$\frac{3.41421356 + 1.82842712 i}{3}$

Rewrite $3.41421356$ as $1 (3.41421356)$ .

$\frac{1 (3.41421356) + 1.82842712 i}{3}$

Factor $1$ out of $1.82842712 i$ .

$\frac{1 (3.41421356) + 1 (1.82842712 i)}{3}$

Factor $1$ out of $1 (3.41421356) + 1 (1.82842712 i)$ .

$\frac{1 (3.41421356 + 1.82842712 i)}{3}$

Factor $3$ out of $3$ .

$\frac{1 (3.41421356 + 1.82842712 i)}{3 (1)}$

Separate fractions.

$\frac{1}{3} \cdot \frac{3.41421356 + 1.82842712 i}{1}$

Divide $1$ by $3$ .

$0.33333333 \frac{3.41421356 + 1.82842712 i}{1}$

Divide $3.41421356 + 1.82842712 i$ by $1$ .

$0.33333333 (3.41421356 + 1.82842712 i)$

Apply the distributive property.

$0.33333333 \cdot 3.41421356 + 0.33333333 (1.82842712 i)$

Multiply $0.33333333$ by $3.41421356$ .

$1.13807118 + 0.33333333 (1.82842712 i)$

Multiply $1.82842712$ by $0.33333333$ .

$1.13807118 + 0.6094757 i$

Write in Standard Form (1+2i)/( square root of 2+i)

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