Problem 45 - Prealgebra

Question

Expand and simplify each of the following. $$\left(-\frac{1}{2}\right)^{2}$$

Step-by-Step Solution

Answer

The expression \( \left(-\frac{1}{2}\right)^{2} \) simplifies to \( \frac{1}{4} \).

Step 1: Understand the Expression

The expression \( \left(-\frac{1}{2}\right)^{2} \) means we need to square the fraction \( -\frac{1}{2} \). This involves multiplying the fraction by itself.

Step 2: Square the Numerator

The numerator of the fraction is \(-1\). When you square it, you calculate \((-1) \times (-1) = 1\). The result is \(1\).

Step 3: Square the Denominator

The denominator of the fraction is \(2\). When you square it, you calculate \(2 \times 2 = 4\). The result is \(4\).

Step 4: Combine the Results

Now that we have squared the numerator and the denominator, we combine them to form the fraction \(\frac{1}{4}\). This is the simplified form of the original expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in prealgebra, allowing students to find equivalent expressions that are easier to work with. When you simplify an expression, you often combine like terms, reduce fractions, or perform other operations to make the expression more straightforward.

In the expression \( \left(-\frac{1}{2}\right)^{2} \), simplifying involves understanding that the negative sign and the division should both be considered when squaring. This example shows how we apply the arithmetic operation of squaring to both parts: the negative sign and the fractional value.

When simplifying expressions like these, remember to handle signs carefully—two negatives result in a positive when multiplied. This will help ensure you follow the right steps in evaluating and reducing expressions.

Squaring Fractions

Squaring a fraction involves multiplying the fraction by itself. This operation applies separately to both the numerator and the denominator.

For the fraction \(-\frac{1}{2}\), squaring means \((-\frac{1}{2}) \times (-\frac{1}{2})\). Let's break it down:

• Square the numerator: \((-1) \times (-1) = 1\). The negative signs cancel each other out, resulting in a positive 1.
• Square the denominator: \(2 \times 2 = 4\). You simply multiply the number by itself.

This results in the fraction \(\frac{1}{4}\), which is more manageable and clearer to interpret.

Basic Arithmetic Operations

Basic arithmetic operations are the foundation of prealgebra and include addition, subtraction, multiplication, and division.

In the context of the expression \(\left(-\frac{1}{2}\right)^{2}\), it highlights the multiplication aspect of arithmetic. Understanding this operation deeply is key to manipulating fractions efficiently.

Multiplication of fractions follows the rule: multiply the numerators together and the denominators together. Negative signs require careful attention, as multiplying two negatives results in a positive.

Mastering these basic arithmetic operations ensures you can tackle more complex algebraic concepts with confidence.