Question

Perform the indicated operation, and write each expression in the standard form \(a+\) bi. $$ 2 i^{4}\left(1+i^{2}\right) $$

Short answer

0

Step-by-step solution

Simplify the powers of i

Recall the powers of the imaginary unit i: - \(i^1 = i\) - \(i^2 = -1\) - \(i^3 = -i\) - \(i^4 = 1\)Hence, \(i^4 = 1\). Replace \(i^4\) with 1 in the expression.\[2i^4(1 + i^2) = 2 \times 1 (1 + i^2)\]

Simplify the term inside the parenthesis

Simplify the expression inside the parenthesis: Recall that \(i^2 = -1\)\[2 \times 1 \times (1 + i^2)=2 \times 1 \times (1 - 1)\]

Simplify the resulting expression

Perform the operation inside the parenthesis first, then multiply:\[2 \times (1 - 1) = 2 \times 0 = 0 \]

Key concepts

Imaginary Unit

The imaginary unit, denoted as i, is a fundamental concept in mathematics, especially in complex numbers. It stems from the need to solve equations that have no real number solutions. The most basic definition of the imaginary unit is: The imaginary unit i is defined as the square root of -1, so \(i=\sqrt{-1}\).
This might sound abstract initially, but it helps in creating a more comprehensive number system. In many mathematical problems and real-life applications, dealing with imaginary numbers becomes crucial. The term 'imaginary' comes from a misconception that these numbers were unreal or fictitious. However, they are as 'real' to mathematics as any real number.

Powers of i

Understanding the powers of i is essential for simplifying complex expressions. Let's go through what happens when i is raised to different powers: \ - \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\)
Each subsequent power of i can be found by continuing this cycle. For instance, \(i^5 = i (i^4 \)) which simplifies to just i. So, \(i^5 = i\).
This cycle of four simplifies calculations significantly. When working with higher powers of i, you only need to determine the remainder when the exponent is divided by 4.

Simplifying Complex Expressions

Simplifying complex expressions involves a few clear steps that often include recognizing patterns, especially those involving the powers of i. Let's use an example to see this in action: Consider the expression \(2i^{4}(1+i^{2}) \)

• First, replace \(i^4\) with 1 (since \(i^4 = 1\)):
  \ 2i^4(1 + i^2) = 2 \times 1 (1 + i^2) \
• Next, simplify inside the parenthesis, knowing that \(i^2 = -1\):
  \2 \times 1 \times (1 - 1) \
  
• Finally, perform the operations inside:
  \2 \times (1 - 1) = 2 \times 0 = 0\.

Following these steps ensures that complex expressions are simplified systematically, resulting in the correct standard form \(a + bi\).