Problem 88 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{2}{3}\right)^{2}$$

Step-by-Step Solution

Answer

The product is \(\frac{4}{9}\).

Step 1: Expand the Exponentiation

We start by solving the exponentiation. When squaring a fraction, we square both the numerator and the denominator separately. This means that: \[\left(\frac{2}{3}\right)^{2} = \frac{2^{2}}{3^{2}}\]

Step 2: Calculate the Squares

Next, we calculate the squares of the numerator and the denominator. \[2^2 = 4 \text{ and } 3^2 = 9\]Thus, the fraction becomes: \[\frac{4}{9}\]

Step 3: Simplify the Fraction

Check if the fraction \(\frac{4}{9}\) can be simplified further. In this case, 4 and 9 have no common factors other than 1, so the fraction is already in its simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential mathematical skill that helps to express a fraction in its simplest form. To simplify a fraction, you must divide both the numerator and the denominator by their greatest common divisor (GCD). The goal is to make the fraction as simple as possible without changing its value.
A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. For example, in the fraction \(\frac{4}{9}\), the greatest common factor of 4 and 9 is 1. Therefore, this fraction is already simplified.
Key steps to simplify a fraction include:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Check the fraction again to ensure it cannot be simplified any further.

Understanding how to simplify fractions ensures clarity and efficiency in mathematical computations.

Exponentiation of Fractions

Exponentiation in fractions involves raising both the numerator and the denominator to a power. For the fraction \(\left(\frac{2}{3}\right)^{2}\), this means raising 2 and 3 separately to the power of 2. The calculation becomes \(\frac{2^{2}}{3^{2}}\).
Let's break down this process:

• Square the numerator: \(2^{2} = 4\).
• Square the denominator: \(3^{2} = 9\).

So, when exponentiating fractions, each component - numerator and denominator - is treated independently. This makes it easier to handle complex expressions, ensuring that each part of the fraction is correctly raised to the power indicated. Remember, always perform operations on the numerator and the denominator separately before moving on to simplification.

Numerator and Denominator

To correctly understand fraction multiplication and operations, it's crucial to grasp the roles of the numerator and the denominator. A fraction is composed of two parts: the numerator, which is above the line, and the denominator, which is below the line.

• The **numerator** indicates how many parts we have.
• The **denominator** reveals how many parts the whole is divided into.

For instance, in \(\frac{2}{3}\), 2 is the numerator indicating two parts, and 3 is the denominator indicating the whole is divided into three parts.
When multiplying fractions, always multiply the numerators together and the denominators together. This is an essential foundation for performing any arithmetic operations involving fractions, as it maintains the proportionality of the fraction.