Problem 92 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{1}{2}\right)^{2}$$

Step-by-Step Solution

Answer

The product is \( \frac{1}{4} \).

Step 1: Understanding the problem

We need to compute the square of the fraction \( \frac{1}{2} \). This means we multiply \( \frac{1}{2} \) by itself.

Step 2: Multiply the fractions

Calculate \( \left(\frac{1}{2}\right)^2 \) by multiplying \( \frac{1}{2} \times \frac{1}{2} \). Multiply the numerators: \( 1 \times 1 = 1 \), and multiply the denominators: \( 2 \times 2 = 4 \). The result is \( \frac{1}{4} \).

Step 3: Simplify the result

Since the fraction \( \frac{1}{4} \) is already in its simplest form, there is no need for further simplification.

Key Concepts

Simplifying Fractions

Simplifying fractions is all about making a fraction as simple as possible, without changing its value. This means reducing the fraction to its lowest terms, so it’s easier to understand and work with.
To simplify a fraction, you should find the greatest common divisor (GCD) of the numerator and denominator and divide both by that number. For example, in the fraction \( \frac{6}{8} \), both the numerator (6) and the denominator (8) can be divided by 2, their GCD, resulting in \( \frac{3}{4} \).
Here are some quick tips for simplifying fractions:

• Identify the largest number that can evenly divide both the numerator and the denominator.
• Remember, the fraction remains the same even if it looks different, because you’re dividing by 1 in a sense (like \( \frac{2}{2}\)).
• If you end up with a fraction like \( \frac{1}{4} \) and can't think of a common divisor other than 1, your fraction is already in simplest form.

Simplified fractions make calculations less cumbersome and results easier to interpret.

Squaring Fractions

Squaring a fraction means multiplying the fraction by itself. It follows the same principles as squaring a whole number
When squaring a fraction, like \( \left(\frac{1}{2}\right)^2 \), you multiply the numerator by itself to get a new numerator, and do the same for the denominator.
For instance, \( \frac{1}{2} \times \frac{1}{2} \) involves:

• Multiplying the numerators: \( 1 \times 1 = 1 \)
• Multiplying the denominators: \( 2 \times 2 = 4 \)
• Resulting in the fraction: \( \frac{1}{4} \)

This result is already in its simplest form, so no further simplification is needed.

Basic Arithmetic Operations

Basic arithmetic operations form the foundation of more complex math topics and are crucial when working with fractions. These include addition, subtraction, multiplication, and division.
When handling fractions, multiplication is straightforward:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the result if possible.

Addition and subtraction require identical denominators, making them slightly trickier. If denominators differ, you'll need a common denominator before proceeding.
Division of fractions involves multiplying by the reciprocal. For example, dividing by \( \frac{3}{4} \) is the same as multiplying by \( \frac{4}{3} \).
Understanding these basic operations helps in effectively solving fraction problems like squaring or simplifying them, reinforcing arithmetic skills all around.