Problem 32 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (2 i)^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to 16.

Step 1: Understand the Expression

The expression is \((2i)^4\). We need to evaluate this complex number raised to the fourth power.

Step 2: Expand the Power

First, calculate \((2i)^4 = (2i) \times (2i) \times (2i) \times (2i)\). This is equivalent to multiplying \(2i\) by itself four times.

Step 3: Simplify Incrementally

Calculate \((2i) \times (2i)\) first. This is equal to \((2^2) \times (i^2) = 4 \times (-1) = -4\), since \(i^2 = -1\).

Step 4: Continue Simplifying

Next, calculate \((-4) \times (2i)\). This equals \(-4 \times 2 \times i = -8i\).

Step 5: Final Calculation

Finally, multiply \(-8i\) by \(2i\). This is \(-8i \times 2i = -16i^2\). Since \(i^2 = -1\), we find that \(-16i^2 = -16 \times (-1) = 16\).

Step 6: Write in Standard Form

The result is a real number, which in the form \(a + bi\) is \(16 + 0i\).

Key Concepts

Powers of Complex Numbers

Complex numbers can be raised to any power, just like real numbers. When we talk about raising a complex number to a power, we often have expressions that involve the imaginary unit, denoted as \(i\).
Understanding powers of complex numbers is crucial as it reveals patterns within the arithmetic properties of complex numbers.
When dealing with powers, specifically, a certain set of steps helps make the operations clear. For example, with the expression \((2i)^4\), there are specific rules and intuitions that guide us through its evaluation.

• The first step is to express the power as a repeated multiplication of the base. In this case, \((2i)^4\) becomes \((2i) \times (2i) \times (2i) \times (2i)\).
• By systematically simplifying pairwise, you eventually simplify the entire expression.

This process highlights how exploratory and intricate powers of complex numbers can be, offering insight into their unique nature.

Imaginary Unit

The imaginary unit, represented by \(i\), is fundamental in the world of complex numbers. It is defined by the property that \(i^2 = -1\).
This definition is what distinguishes it from real numbers and enables complex numbers to represent values that real numbers cannot.
When handling expressions that include the imaginary unit, such as \((2i)^2\), it's crucial to remember that multiplying by \(i^2\) results in -1.

• This property plays a central role when working through powers of complex numbers, as each pair of \(i\) will collectively contribute a factor of \(-1\) to the overall product.
• The relation \(i^2 = -1\) essentially transforms seemingly abstract expressions into tangible solutions.

The imaginary unit is, therefore, a powerful tool that opens up a new dimension of solving equations and analyzing solutions.

Simplification of Expressions

Simplification in mathematics is about making an expression easier to understand and work with, by transforming it into a simpler or more concise form.
When simplifying expressions involving complex numbers, the goal is often to express them in the standard form \(a + bi\), where \(a\) and \(b\) are real numbers.
During each stage, ensure you perform the arithmetic correctly, especially when negative signs and the imaginary unit are involved.

• A key part of simplification is recognizing and applying the rule \(i^2 = -1\) accurately. This often allows terms to be combined or cancelled out, bringing the expression to a reduced form.
• Moreover, simplifying power expressions, like \((2i)^4 = 16 + 0i\), involves recognizing when you've fully collapsed it into a form without remaining powers of \(i\).

Simplification is an absolute necessity not only for solving but also for providing intuitive understanding of complex number operations.