Problem 44 - Prealgebra

Question

Simplify each expression. $$\left(1+\frac{3}{8}\right)\left(1-\frac{3}{8}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{55}{64}\).

Step 1: Recognize the Expression Form

Identify the expression \( \left(1+\frac{3}{8}\right)\left(1-\frac{3}{8}\right) \) as a product of a sum and a difference. This is in the form of \((a+b)(a-b)\), where \(a = 1\) and \(b = \frac{3}{8}\).

Step 2: Apply the Difference of Squares Formula

Use the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). In this expression, \(a = 1\) and \(b = \frac{3}{8}\), so apply the formula as follows: \(1^2 - \left(\frac{3}{8}\right)^2\).

Step 3: Simplify the Squares

Calculate \(1^2\), which equals 1, and \(\left(\frac{3}{8}\right)^2\), which equals \(\frac{9}{64}\). Substitute these values back into the expression from Step 2: \(1 - \frac{9}{64}\).

Step 4: Subtract the Fractions

Write 1 as a fraction with the same denominator as \(\frac{9}{64}\). This converts 1 to \(\frac{64}{64}\). Now we can subtract: \(\frac{64}{64} - \frac{9}{64} = \frac{55}{64}\).

Step 5: Conclude Solution

Thus, the simplified form of \( \left(1+\frac{3}{8}\right)\left(1-\frac{3}{8}\right) \) is \(\frac{55}{64}\).

Key Concepts

Understanding the Difference of Squares

One of the elegant shortcuts in algebra is understanding the difference of squares, represented by the expression \((a+b)(a-b) = a^2 - b^2\). This formula is particularly useful when simplifying expressions that involve a sum and a difference of the same two terms.
This expression type appears frequently, and recognizing it can significantly speed up calculations. It eliminates the need to expand both binomials completely, which can save time and reduce complexity in your calculations.

• In our example, \((1 + \frac{3}{8})(1 - \frac{3}{8})\) fits perfectly into this template with \(a=1\) and \(b=\frac{3}{8}\).
• We substitute these values into the formula, yielding \(1^2 - \left(\frac{3}{8}\right)^2\).

This compact expression allows us to simplify directly, leveraging basic arithmetic operations.

Simplifying with Fraction Subtraction

Fraction subtraction might seem daunting at first, but once you know the trick, it becomes straightforward. A crucial step in our solution is rewriting a whole number to have the same denominator as the fraction it is paired with.
This pivots the expression into a more manageable form for subtraction.

• For instance, converting 1 into \(\frac{64}{64}\) helps in directly subtracting \(\frac{9}{64}\) from it.
• Once the denominators are the same, you can straightforwardly subtract the numerators: \(64 - 9 = 55\), giving you the final result \(\frac{55}{64}\).

This process underlines the simplicity of handling fractional components when they share a common denominator.

Deciphering Mathematical Formulas

Mathematical formulas are powerful tools that provide a direct path to solve complex problems. Understanding how to decipher and apply them can boost your efficiency in problem-solving.
These formulas capture rich concepts in simple terms. Recognizing a formula's applicability is as crucial as performing arithmetic calculations.

• The difference of squares formula \((a+b)(a-b) = a^2 - b^2\) is a classic example. It allows us to transform polynomial expressions quickly into simpler forms.
• In our problem, it reduced a potentially lengthy distribution process into a single subtraction operation.

Grasping the mental map for how these formulas fit into different mathematical settings strengthens your foundational skills and builds your confidence in tackling various types of mathematical challenges.