Problem 46 - Prealgebra

Question

Expand and simplify each of the following. $$\left(-\frac{1}{3}\right)^{2}$$

Step-by-Step Solution

Answer

The expanded and simplified form is \(\frac{1}{9}\).

Step 1: Understand the Exercise

The exercise asks us to expand and simplify the expression \((-\frac{1}{3})^2\). This involves calculating the square of a negative fraction.

Step 2: Rewrite the Expression

We rewrite \((-\frac{1}{3})^2\) as \((-\frac{1}{3})\times (-\frac{1}{3})\) to express squaring as multiplication.

Step 3: Multiply the Fractions

Now, we multiply \(-\frac{1}{3}\) by itself: \(-1\times -1 = 1\) for the numerators, and \(3\times 3 = 9\) for the denominators. Thus, \((-\frac{1}{3})\times (-\frac{1}{3}) = \frac{1}{9}\).

Step 4: Simplify the Result

The result \(\frac{1}{9}\) is already in its simplest form, since the fraction cannot be reduced further.

Key Concepts

Understanding Negative Numbers

Negative numbers can seem a bit confusing at first, but they're simply numbers that are less than zero. They are usually denoted with a minus sign (−) in front of them.
A common property of negative numbers is how they behave during multiplication and division.
When you multiply two negative numbers together, the result is always a positive number. This might be surprising, but here’s why: multiplying a negative by a negative essentially "flips" it back to positive.

• A negative times a negative equals a positive (e.g., \(-2 \times -3 = 6\)).
• A positive times a negative equals a negative (e.g., \(3 \times -2 = -6\)).
• These rules are crucial when dealing with exponents, particularly when a negative is raised to an even power, like squaring \(-\left(\frac{1}{3}\right)\).

Exploring Fractions

Fractions are a way of expressing numbers that are not whole, created out of a numerator and a denominator separated by a division line. They present values as parts of a whole.
For example, \(\frac{1}{3}\) means one third of a whole.
The concept of fractions can also combine with negative numbers, as in \(-\frac{1}{3}\), which indicates a negative fraction.

• The numerator is the top number of the fraction, representing how many parts are selected.
• The denominator is the bottom number of the fraction and shows how many parts make up a whole.
• Fractions can be negative if either the numerator or denominator is negative, but not both simultaneously because \(-\left\(\frac{a}{b}\right\) = \frac{-a}{b} = \frac{a}{-b}\).

The Multiplication of Fractions

Multiplying fractions involves a simple yet specific process: multiply the numerators together for a new numerator, and multiply the denominators together for a new denominator. Let's break this down:
To multiply \(-\frac{1}{3} \times -\frac{1}{3}\), we take:

• The numerators: \(-1 \times -1 = 1\).
• The denominators: \(3 \times 3 = 9\).

Given this operation, our result is \(\frac{1}{9}\).
While performing these calculations, remember the property that two negatives make a positive, crucial for both the numerators and denominators in this case.

Simplifying Expressions to Their Core

Simplifying expressions is the process of reducing them to their simplest form. This often involves various operations to ensure you have the most compact expression possible.
For fractions, simplification means ensuring the numerator and denominator have no common factors other than 1.

• The expression \(\frac{1}{9}\) is already simplified since 1 and 9 have no common factors.
• Simplifying helps in solving problems faster and reduces the complexity of calculations.
• Always check if the fraction parts can be divided by a common number to reduce them further; here, it is already the simplest form.

Simplifying helps both in solving mathematical problems and in clear communication of the results.