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

Solution:

Step:1

To rationalize the denominator, multiply both the numerator and denominator of $\frac{1+2i}{\sqrt{2}+i}$ by the conjugate of the denominator: $\frac{\sqrt{2}-i}{\sqrt{2}-i}$.

Step:2

Perform the multiplication.

Step:2.1

Combine the terms: $\frac{(1+2i)(\sqrt{2}-i)}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2

Expand the numerator.

Step:2.2.1

Use the FOIL method to expand $(1+2i)(\sqrt{2}-i)$.

Step:2.2.1.1

Distribute each term: $\frac{1(\sqrt{2}-i)+2i(\sqrt{2}-i)}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.1.2

Continue distribution: $\frac{1\cdot \sqrt{2}+1(-i)+2i(\sqrt{2}-i)}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.1.3

Finish distribution: $\frac{1\cdot \sqrt{2}+1(-i)+2i\cdot \sqrt{2}+2i(-i)}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.2

Combine like terms in the numerator.

Step:2.2.2.1

Simplify each term.

Step:2.2.2.1.1

Multiply $\sqrt{2}$ by $1$: $\frac{\sqrt{2}+1(-i)+2i\cdot \sqrt{2}+2i(-i)}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.2.1.2

Multiply $-i$ by $1$: $\frac{\sqrt{2}-i+2i\cdot \sqrt{2}+2i(-i)}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.2.1.3

Multiply $\sqrt{2}$ by $2i$: $\frac{\sqrt{2}-i+2\sqrt{2}i+2i(-i)}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.2.1.4

Multiply $2i$ by $-i$.

Step:2.2.2.1.4.1

Multiply $-1$ by $2i$: $\frac{\sqrt{2}-i+2\sqrt{2}i-2i^2}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.2.1.4.2

Recognize that $i^1\cdot i^1=i^2$.

Step:2.2.2.1.4.3

Apply the power rule $a^m\cdot a^n=a^{m+n}$ to combine exponents: $\frac{\sqrt{2}-i+2\sqrt{2}i-2i^2}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.2.2.1.4.4

Recognize that $i^2=-1$ and simplify: $\frac{\sqrt{2}-i+2\sqrt{2}i+2}{(\sqrt{2}+i)(\sqrt{2}-i)}$.

Step:2.3

Simplify the denominator.

Step:2.3.1

Use the FOIL method to expand $(\sqrt{2}+i)(\sqrt{2}-i)$.

Step:2.3.1.1

Apply distribution: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{\sqrt{2}(\sqrt{2}-i)+i(\sqrt{2}-i)}$.

Step:2.3.1.2

Continue distribution: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{\sqrt{2}\cdot \sqrt{2}+\sqrt{2}(-i)+i(\sqrt{2}-i)}$.

Step:2.3.1.3

Finish distribution: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{\sqrt{2}\cdot \sqrt{2}+\sqrt{2}(-i)+i\cdot \sqrt{2}+i(-i)}$.

Step:2.3.2

Combine like terms in the denominator.

Step:2.3.2.1

Multiply $\sqrt{2}$ by $\sqrt{2}$: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{2+\sqrt{2}(-i)+i\cdot \sqrt{2}+i(-i)}$.

Step:2.3.2.2

Multiply $-i$ by $\sqrt{2}$: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{2-\sqrt{2}i+i\cdot \sqrt{2}+i(-i)}$.

Step:2.3.2.3

Multiply $i$ by $\sqrt{2}$: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{2-\sqrt{2}i+\sqrt{2}i-i^2}$.

Step:2.3.2.4

Recognize that $i^2=-1$ and simplify: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{2-\sqrt{2}i+\sqrt{2}i+1}$.

Step:2.3.3

Combine like terms: $\frac{\sqrt{2}+2\sqrt{2}i-i+2}{3}$.

Step:3

Divide each term in the numerator by the denominator: $\frac{\sqrt{2}}{3}+\frac{2\sqrt{2}i}{3}-\frac{i}{3}+\frac{2}{3}$.

Step:4

Combine the real parts and the imaginary parts: $(\frac{\sqrt{2}}{3}+\frac{2}{3})+(\frac{2\sqrt{2}}{3}-\frac{1}{3})i$.

Step:5

Simplify the expression: $\frac{3\sqrt{2}+2}{3}+\frac{2\sqrt{2}-1}{3}i$.

Step:6

Write the final answer in standard form: $a+bi$ where $a=\frac{3\sqrt{2}+2}{3}$ and $b=\frac{2\sqrt{2}-1}{3}$.

Knowledge Notes:

To write a complex number in standard form after division, we need to eliminate the imaginary unit $i$ from the denominator. This is done by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a+bi$ is $a-bi$. Multiplying a complex number by its conjugate results in a real number because $(a+bi)(a-bi)=a^2-b^2{i}^2=a^2+b^2$ (since $i^2=-1$).

The FOIL method stands for First, Outer, Inner, Last and is a technique used to multiply two binomials. The distributive property, also known as the distributive law of multiplication, states that $a(b+c)=ab+ac$. This property is used extensively when expanding expressions.

When simplifying complex numbers, it's important to combine like terms and to remember that $i^2=-1$. This allows us to convert terms with $i^2$ into real numbers and thus simplify the expression further.

The standard form of a complex number is $a+bi$, where $a$ is the real part and $b$ is the imaginary part. After simplifying the expression, we should always aim to write the complex number in this form.