Problem 36 - Algebra and Trigonometry

Question

Perform the indicated operations. \(\left(\frac{1}{2}-\frac{1}{3}\right)\left(\frac{1}{2}+\frac{1}{3}\right)\)

Step-by-Step Solution

Answer

The solution is \(\frac{5}{36}\).

Step 1: Recognize the Expression Form

The given expression is \(\left(\frac{1}{2}-\frac{1}{3}\right)\left(\frac{1}{2}+\frac{1}{3}\right)\)which is in the form of a difference of squares, which is \((a-b)(a+b) = a^2 - b^2\). Here, \(a = \frac{1}{2}\) and \(b = \frac{1}{3}\).

Step 2: Calculate Square of Each Term

First, find the square of \(a\) and \(b\).\[a^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\]\[b^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}\]

Step 3: Apply Difference of Squares Formula

Substitute \(a^2\) and \(b^2\) into the difference of squares formula.\[a^2 - b^2 = \frac{1}{4} - \frac{1}{9}\]

Step 4: Compute the Difference of Fractions

To subtract the fractions, find a common denominator. The denominators are 4 and 9, so the least common denominator is 36. Rewrite the fractions:\[\frac{1}{4} = \frac{9}{36}, \quad \frac{1}{9} = \frac{4}{36}\]

Step 5: Subtract the Fractions

Subtract the fractions:\[\frac{9}{36} - \frac{4}{36} = \frac{5}{36}\]

Step 6: State the Final Answer

The result of the expression \(\left(\frac{1}{2}-\frac{1}{3}\right)\left(\frac{1}{2}+\frac{1}{3}\right)\) is \(\frac{5}{36}\).

Key Concepts

Fractions

Fractions consist of a numerator and a denominator, representing parts of a whole. In the fraction \( \frac{1}{2} \), "1" is the numerator, and "2" is the denominator, indicating one half. Working with fractions involves operations like addition, subtraction, multiplication, and division. A key component is the common denominator, crucial in addition and subtraction.

• When adding or subtracting, find a common denominator, which is usually the least common multiple of the denominators involved.
• Convert both fractions to equivalent fractions with the same denominator.
• Then, perform the addition or subtraction on the numerators, keeping the common denominator unchanged.

In the problem, the fractions \( \frac{1}{4} \) and \( \frac{1}{9} \) became \( \frac{9}{36} \) and \( \frac{4}{36} \) after conversion. This is because 36 is the smallest common multiple of 4 and 9.

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables raised to powers and coefficients. They can vary from simple to complex expressions, some involving only one variable, while others contain multiple. Here, our expression was structured as a product of two binomials: \( \left( \frac{1}{2} - \frac{1}{3} \right) \) and \( \left( \frac{1}{2} + \frac{1}{3} \right) \). Characteristics of Polynomial Expressions:

• They can include both positive and negative terms.
• Exponentiation in polynomials is typically integer-based and non-negative.
• The operations involved are mathematical operations like addition, subtraction, multiplication, etc.

In the context of our task, recognizing the expression form as a difference of squares, a specific type of polynomial factorization, helped simplify the computation significantly.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions, or factors, which when multiplied together give the original expression. It's a cornerstone technique in algebra that simplifies complex expressions.One special case is the difference of squares, a concept that applies when we have two terms squared and subtracted, forming \( a^2 - b^2 = (a-b)(a+b) \).In this exercise, identifying the given expression \( \left( \frac{1}{2} - \frac{1}{3} \right)( \frac{1}{2} + \frac{1}{3} ) \) as a difference of squares allowed us to use the formula directly. By calculating \( a^2 \) and \( b^2 \), we restructured the expression into a simpler form, avoiding expansive multiplication and direct computation. Hence

• Acknowledge the form: Recognizing an expression as a factored form speeds up simplification.
• Apply the appropriate factorization formula for quick results.

Factoring can transform intimidating problems into manageable tasks by revealing the underlying simplicity of an expression.