Problem 36 - Algebra 2 and Trigonometry

Question

In \(35-43,\) write each number in simplest form. $$ 5 i^{2}+2 i^{4} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \(-3\).

Step 1: Simplify the powers of i

The number 'i' represents the imaginary unit, where \(i = \sqrt{-1}\). Therefore, we can simplify the powers of 'i' using these rules: \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). We start by substituting these values into the expression.

Step 2: Substitute the powers into the expression

Replace \(i^2\) and \(i^4\) in the expression \(5i^2 + 2i^4\) with their corresponding values. Hence, we have:\[5(-1) + 2(1)\]

Step 3: Simplify the expression

Now, simplify the expression obtained by performing the substitution:\[-5 + 2 = -3\]

Key Concepts

Imaginary Unit

In mathematics, we often encounter limitations when dealing with the square roots of negative numbers. To overcome this, we use the concept of the **imaginary unit**, symbolized as 'i'.
The imaginary unit is defined as:

• \( i = \sqrt{-1} \)
• It allows us to extend the set of real numbers to include what are known as **complex numbers**.

The introduction of the imaginary unit leads us to new and useful rules for working with powers of 'i', which simplifies manipulation and operations in mathematics.
This is particularly helpful in fields like engineering and physics, where complex numbers are used to model real-world phenomena.

Powers of i

Understanding the powers of 'i' is critical to working efficiently with expressions involving imaginary numbers.
By applying the imaginary unit, we derive specific rules:

• \( i^1 = i \)
• \( i^2 = -1 \) : This is one of the most important properties, showing why 'i' is imaginary.
• \( i^3 = -i \) : This is derived by multiplying \(i^2\) by 'i'.
• \( i^4 = 1 \) : Represents a full cycle of the powers of 'i', as repeating powers follow this cycle.

These cyclical patterns mean that any higher power of 'i' can be reduced using these fundamental rules. For example, to calculate \(i^5\), notice that it equals \(i^{4+1} = i^4 \cdot i^1 = 1 \cdot i = i\).
Recognizing these cycles helps greatly in simplifying expressions.

Simplifying Expressions

Simplifying expressions with imaginary numbers often requires substituting the powers of 'i' with their corresponding values from the cyclical pattern we identified. Here is the step-by-step approach using the example given:Start with the expression \(5i^2 + 2i^4\).
Recognize the powers and replace them based on their reduced forms:

• Since \(i^2 = -1\), replace \(i^2\) with \(-1\).
• Since \(i^4 = 1\), substitute \(i^4\) with \(1\).

This substitution gives the expression:

• \(5(-1) + 2(1)\)

Now, simplify by performing typical arithmetic operations:

• Calculate \(5 \times -1 = -5\)
• Calculate \(2 \times 1 = 2\)
• Add the results: \(-5 + 2 = -3\)

Thus, the simplified result of the expression is \(-3\).
Such simplifications can transform complex-looking expressions into much simpler forms, making them easier to handle in calculations and problem solving.