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: $\left(\frac{\sqrt{2}}{3} + \frac{2}{3}\right) + \left(\frac{2\sqrt{2}}{3} - \frac{1}{3}\right)i$.

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 $(a + bi)(a - bi) = a^2 - b^2i^2 = a^2 + b^2$ (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.