Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the complex number and write it in standard form. $$(\sqrt{-2})^{6}$$

Step-by-Step Solution

Answer

The simplified form of the complex number is -8.

Step 1: Understanding the notation of complex numbers

First, we need to understand the notation. In standard form, complex numbers are written as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property that \(i^{2} = -1\). The expression \(\sqrt{-2}\) is an example of a complex number, where \(a = 0\) and \(b = \sqrt{2}\). So, we can write \(\sqrt{-2} = \sqrt{2} \cdot i\).

Step 2: Simplifying the power

Next, we need to calculate \((\sqrt{2} \cdot i)^{6}\). This can be expressed as (\(\sqrt{2}^{6}\)) and \(i^{6}\), according to the power of a product property. Simplifying this gives us \(2^{3} \cdot i^{6} = 8 \cdot i^{6}\).

Step 3: Dealing with high powers of \(i\)

In complex numbers, powers of \(i\) can be simplified using the property that \(i^{4n} = 1\) for any integer \(n\). This means \(i^{6} = i^{4} \cdot i^{2} = 1 \cdot (-1) = -1\). So, \(8 \cdot i^{6}\) simplifies to \(8 \cdot (-1) = -8\). This is a real number.