Problem 41 - Prealgebra

Question

Expand and simplify each of the following. $$\left(\frac{2}{3}\right)^{2}$$

Step-by-Step Solution

Answer

The expanded and simplified result is \(\frac{4}{9}\).

Step 1: Understand the Power

The expression \(\left(\frac{2}{3}\right)^{2}\) requires you to raise \(\frac{2}{3}\) to the power of 2. This means you'll multiply \(\frac{2}{3}\) by itself.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \(2 \times 2 = 4\).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \(3 \times 3 = 9\).

Step 4: Write the Result as a Fraction

The result of the operation \(\left(\frac{2}{3}\right)^{2}\) is \(\frac{4}{9}\) because you have \(4\) as the numerator and \(9\) as the denominator after the multiplications.

Key Concepts

Understanding Fractions in Exponentiation

When dealing with fractions in exponentiation, it is important to remember that a fraction consists of two parts: the numerator and the denominator. In the expression \(\left(\frac{2}{3}\right)^2\), the number \(2\) in the fraction \(\frac{2}{3}\) is the numerator, and \(3\) is the denominator.

Exponentiation involving fractions means that both the numerator and the denominator must be raised to the power specified. This is a crucial step in ensuring that each part of the fraction is handled correctly.

• When you see an exponent, like \(2\) here, apply it separately to both the numerator and the denominator.
• Multiplying the fraction by itself means you'll calculate \(\left(2\times2\right)\) for the numerator and \(\left(3\times3\right)\) for the denominator.

Understanding the components of a fraction helps in executing these operations with precision.

Numerators and Denominators in Fractions

Fractions are made up of numerators and denominators, which serve specific purposes in mathematical expressions. Here's a breakdown of their roles:

The **Numerator** is the top number in a fraction and represents how many parts of the whole are being considered. For example, in \(\frac{2}{3}\), the number \(2\) is the numerator. In multiplication, you treat numerators just like whole numbers. So, when raising a fraction to a power, you multiply the numerator by itself as specified by the exponent.

The **Denominator** is the bottom number and indicates the total number of equal parts in the whole. In our example, \(3\) is the denominator. When raising a fraction to a power, multiply the denominator by itself as specified by the exponent as well.

• Consistency is key; what you do to the numerator, you must do to the denominator.
• Always consider both components separately before re-uniting them into a single fraction.

Knowing the proper handling of numerators and denominators will guide you through more complex fraction problems.

Simplifying Expressions Involving Fractions

Simplifying expressions is a vital step in making your math solutions more elegant and understandable. When you raise fractions to powers, you'll tend to end up with results that can appear more complex than necessary.

Simplifying involves reducing fractions to their lowest terms, ensuring they are easy to interpret and work with in further calculations. To simplify, follow these steps:

• Find the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both by the GCD to reduce the fraction to its simplest form.

In the original problem \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\), there's no need to simplify further because \(4\) and \(9\) have no common divisors other than 1.

Proper simplification results in clearer, more efficient mathematics, which is especially beneficial in higher-level equations and when comparing or combining fractions.