Math Snap

STEP 1

1. We are given a complex number in the form of a fraction.
2. The goal is to express this complex number in the standard form $a + bi$.
3. To achieve this, we will multiply the numerator and the denominator by the conjugate of the denominator.

STEP 2

1. Identify and multiply by the conjugate of the denominator.
2. Simplify the expression to obtain the form $a + bi$.

STEP 3

Identify the conjugate of the denominator $\sqrt{2} + \sqrt{2}i$. The conjugate is $\sqrt{2} - \sqrt{2}i$.
Multiply both the numerator and the denominator by this conjugate:
$$\frac{1}{\sqrt{2}+\sqrt{2} i} \times \frac{\sqrt{2} - \sqrt{2} i}{\sqrt{2} - \sqrt{2} i}$$

STEP 4

Simplify the denominator using the difference of squares formula:
$$(\sqrt{2} + \sqrt{2}i)(\sqrt{2} - \sqrt{2}i) = (\sqrt{2})^2 - (\sqrt{2}i)^2$$ Calculate each term:
$$(\sqrt{2})^2 = 2 \quad \text{and} \quad (\sqrt{2}i)^2 = 2i^2 = -2$$ Thus, the denominator becomes:
$$2 - (-2) = 2 + 2 = 4$$

STEP 5

Simplify the numerator:
$$1 \times (\sqrt{2} - \sqrt{2}i) = \sqrt{2} - \sqrt{2}i$$

Combine the results to express the fraction in the form $a + bi$:
$$\frac{\sqrt{2} - \sqrt{2}i}{4} = \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{4}i$$ Thus, the expression in the form $a + bi$ is:
$$a = \frac{\sqrt{2}}{4}, \quad b = -\frac{\sqrt{2}}{4}$$