Problem 14 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. [Examples 1–3] $$\left(2-\frac{1}{4}\right)\left(2+\frac{1}{4}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{63}{16} \).

Step 1: Understanding the Expression

The expression given is \( \left(2-\frac{1}{4}\right)\left(2+\frac{1}{4}\right) \). This is a mathematical expression involving parentheses and fractions. It is an application of the difference of squares formula.

Step 2: Recognize the Pattern

The expression \( \left(2-\frac{1}{4}\right)\left(2+\frac{1}{4}\right) \) matches the pattern \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 2\) and \(b = \frac{1}{4}\).

Step 3: Apply the Difference of Squares Formula

Using the formula \((a-b)(a+b) = a^2 - b^2\), we substitute \(a\) and \(b\) with 2 and \(\frac{1}{4}\) respectively: \[ a^2 - b^2 = 2^2 - \left(\frac{1}{4}\right)^2 \]

Step 4: Calculate \(a^2\) and \(b^2\)

Calculate \(a^2\) as \(2^2 = 4\) and \(b^2\) as \(\left(\frac{1}{4}\right)^2 = \frac{1}{16}\).

Step 5: Subtract \(b^2\) from \(a^2\)

Subtract \(\frac{1}{16}\) from 4 to simplify:\[ 4 - \frac{1}{16} \] To do this, convert 4 to a fraction with the same denominator: \(4 = \frac{64}{16}\). Then, subtract:\[ \frac{64}{16} - \frac{1}{16} = \frac{63}{16} \]

Step 6: Present the Simplified Result

The simplified form of the expression \( \left(2-\frac{1}{4}\right)\left(2+\frac{1}{4}\right) \) is \( \frac{63}{16} \).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic identity that simplifies expressions in the form

• (a-b)(a+b) = a^2 - b^2

This pattern arises because when you multiply two binomials, the middle terms cancel each other out, making it particularly handy for simplifying complex expressions.
In this exercise, our expression \[(2-\frac{1}{4})(2+\frac{1}{4})\]exactly fits the form of the difference of squares.

• Here, \(a = 2\) and \(b = \frac{1}{4}\).
• When using the formula, simply calculate \(a^2\) and \(b^2\) individually.

The power of this identity lies in its ability to quickly simplify expressions, saving both time and effort in solving mathematical problems.

Fractions

Fractions can sometimes seem intimidating, but understanding them is crucial. Fractions represent a part of a whole and consist of a numerator (top number) and a denominator (bottom number). In our example, we encounter the fraction \[\frac{1}{4}\]where 1 is the numerator and 4 is the denominator.
To work with fractions effectively:

• Ensure the denominators are equal before performing addition or subtraction.
• Convert whole numbers into fractions when they need to have a common base with other fraction numbers.

For instance, before subtracting fractions, you might need to turn a whole number like 4 into \[\frac{64}{16}\]so that it shares the same base as the fraction you're working with. Understanding how to maneuver around fractions can transform a daunting task into a straightforward one.

Simplifying Expressions

Simplifying expressions involves reducing them to their most compact and understandable form, which is often necessary for easier interpretation or further calculations. In our example, we simplified \[\left(2-\frac{1}{4}\right)\left(2+\frac{1}{4}\right)\]using:

• The difference of squares to break down the expression efficiently.
• Arithmetic operations on fractions to combine terms correctly.

To simplify, follow these steps:

• Apply relevant algebraic identities to reduce the expression first.
• Make sure to perform calculations correctly by converting necessary values into compatible forms, like fractions with the same denominator.
• Check your work by ensuring that each part of the expression has been refined and essential operations have been double-checked.

By mastering these techniques, you can tackle even the most complex expressions with confidence.