Problem 66 - Algebra 1

Question

Simplify the expression. $$\frac{i}{-4}(-4)$$

Step-by-Step Solution

Answer

The simplified expression is \(i\).

Step 1: Understand the problem

The problem asks to simplify the expression \(\frac{i}{-4}(-4)\). Knowing the definition of the imaginary unit \(i\) is the foundation for solving the problem from here.

Step 2: Multiply \(\frac{i}{-4}\) by -4

According to the order of operations (BIDMAS/BODMAS/PEDMAS), any calculation inside brackets has the highest precedence. Hence, we first need to process the calculation inside the brackets. \(\frac{i}{-4}(-4) = i\) due to -4 in the numerator and denominator cancelling each other out.

Step 3: Final simplification

Since the multiplication operation leads to i only, there are no further simplification steps required. Thus, the expression is already in its simplest form.

Key Concepts

Imaginary Unit

The concept of the imaginary unit is foundational in complex number theory. An imaginary unit, denoted as \( i \), represents the mathematical expression for the square root of -1. This might seem perplexing at first, because when we deal with real numbers only, there's no number that, when squared, gives a negative result. In the realm of complex numbers, however, \( i \) is essential and is defined such that \( i^2 = -1 \). When working with imaginary units in algebra, remember that they follow certain rules: \( i^3 = i^2 \cdot i = -i \), \( i^4 = 1 \), and so on, in a cyclical pattern. In algebraic operations, treating \( i \) correctly is crucial for simplifying expressions involving complex numbers.

Order of Operations

To simplify mathematical expressions correctly, it's vital to adhere to the order of operations. Commonly known through acronyms such as BIDMAS, BODMAS, or PEMDAS, the order of operations dictates the sequence to follow during simplification: Brackets/Parentheses, Indices/Exponents/Orders (such as powers and roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). In our example, \( \frac{i}{-4}(-4) \), the expression within the parenthesis (-4) is considered first. By following these rules, particularly the application of the multiplication operation after addressing division, we ensure that expressions are simplified in a logically consistent manner, which avoids errors in calculations involving multiple terms and operations.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. This involves combining like terms, factoring, expanding expressions, and canceling out terms when possible. In the expression \( \frac{i}{-4}(-4) \), simplification involves canceling out the -4 in the denominator with the -4 in the numerator. Since the numerator and the denominator are the same (except for their sign), they cancel each other out, leaving just the numerator's \( i \). This is because any number divided by itself is 1, but in this case, the -4 also negates the negative in the denominator, simplifying to just \( i \), which is the simplest form of the expression. Understanding algebraic simplification principles is crucial in not just solving textbook problems, but also in simplifying complex equations in advanced mathematics and applied sciences.