Problem 31 - Algebra 2

Question

Simplify. $$ 3 i(-5 i)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(-75i\).

Step 1: Understand the Problem

We need to simplify the expression \(3 i(-5i)^2\). This involves performing operations on the imaginary unit \(i\) and applying the laws of exponents.

Step 2: Evaluate the Exponent

First, we need to evaluate \((-5i)^2\). We know that \((a b)^2 = a^2 b^2\), so \((-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot i^2\). Since \(i^2 = -1\), we have \(25 \cdot (-1) = -25\).

Step 3: Multiply the Remaining Terms

Now, substitute \((-5i)^2\) with \(-25\) into the original expression: \(3 i (-25)\). Multiply the terms: \(3 i \cdot (-25) = -75i\).

Step 4: Final Simplification

There are no further terms to simplify. Thus, the expression simplifies to \(-75i\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit, denoted as \(i\), is a core idea in complex numbers. The main property of \(i\) is that \(i^2 = -1\). This simple yet powerful definition allows us to solve equations that have no real solutions, like \(x^2 + 1 = 0\). Without the imaginary unit, such equations would remain unsolvable under real numbers. In the realm of complex numbers, \(i\) is used to extend the traditional number line into a plane, allowing the expression and manipulation of numbers beyond the real dimension. Understanding \(i\) is essential when dealing with complex arithmetic, as it serves as the foundation for all operations involving complex numbers, especially when taking powers of imaginary units.

Laws of Exponents

Mastering the laws of exponents is crucial when working with complex numbers. These laws help us efficiently simplify expressions that involve powers. The general rules to remember are:

• \(a^m \cdot a^n = a^{m+n}\)
• \((a^m)^n = a^{m \cdot n}\)
• \((a \cdot b)^n = a^n \cdot b^n\)

In the context of the provided exercise, the expression \((-5i)^2\) is simplified by using these rules. First, we separate the expression into factors: - \((-5)^2\)- \(i^2\)This allows us to simplify by calculating each part individually before combining the results. By applying the laws of exponents, we effectively manage complex expressions in a clear and systematic way.

Simplification of Expressions

Simplifying expressions involving complex numbers often requires careful application of fundamental principles. When we encounter terms like \((-5i)^2\), we must break them down correctly. Simplification involves:

• Identifying and breaking down terms using known properties, like \(i^2 = -1\).
• Applying arithmetic operations step-by-step, focusing on order of operations, such as handling powers before multiplication.
• Replacing complex terms with simpler equivalents, such as transforming \(i^2\) into \(-1\).

Through simplification, expressions become more straightforward, facilitating easier computation and better understanding. This process is critical when following exercises that initially seem complicated but unravel upon systematic analysis.

Multiplication of Complex Numbers

To multiply complex numbers, one must consider both the real and imaginary components. The operation is akin to multiplying polynomials: distribute terms and simplify. In the given example, we simplify by:

• Identifying and multiplying coefficients, like combining \(3\) and \(-25\).
• Ensuring imaginary units are treated correctly, applying properties like \(i^2 = -1\).
• Combining like terms, keeping results in the simplest form.

The result of multiplying complex numbers either involves pure imaginary numbers, like \(-75i\), as in the example, or a combination of real and imaginary parts. This systematic approach helps ensure accurate computation, leading to correct results.