Problem 44 - Prealgebra

Question

Expand and simplify each of the following. $$\left(-\frac{2}{7}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{4}{49}\).

Step 1: Understand the Basics

The expression we need to simplify is \( \left( -\frac{2}{7} \right)^2 \). It consists of a fraction raised to the power of 2. When raising a fraction to a power, both the numerator and the denominator need to be raised to that power. Since the fraction is negative, squaring it will make it positive.

Step 2: Square the Numerator and Denominator Separately

First, focus on the numerator. The numerator of \( -\frac{2}{7} \) is -2. Squaring the numerator: \((-2)^2 = 4\). Next, focus on the denominator. The denominator is 7. Squaring the denominator: \(7^2 = 49\).

Step 3: Combine the Results

Combine your results from squaring the numerator and the denominator. The new fraction is created by putting \(4\) over \(49\):\[\left(\frac{-2}{7}\right)^2 = \frac{4}{49} \] This is the expanded and simplified form of the given expression.

Step 4: Concluding the Calculation

The negative sign becomes irrelevant because "\((-1)^2\)" is always positive. Therefore, the final answer is positive \(\frac{4}{49}\).

Key Concepts

Understanding Exponents

Exponents are a shorthand way of saying you are multiplying a number by itself a certain number of times. In mathematics, an exponent is written as a small number placed to the upper right of the base number. For example, in the expression \( x^n \), \( x \) is the base and \( n \) is the exponent, which tells you how many times \( x \) is multiplied by itself.

• If \( n = 2 \), it means the base is squared.
• If \( n = 3 \), it means the base is cubed.

When you have a negative number as the base, like \( -rac{2}{7} \), squaring it means you'll multiply it by itself twice. Regardless of how many negatives are used, any even exponent (like 2, 4, 6, etc.) will result in a positive product. This is because negative times negative results in a positive.

Working with Fractions

Fractions represent a part of a whole or a division of numbers. Each fraction consists of a numerator, the top part of the fraction, and a denominator, the bottom part. The general format is \( \frac{numerator}{denominator} \).

• The numerator tells you how many parts you have.
• The denominator tells you into how many parts the whole is divided.

When squaring a fraction such as \(-\left(\frac{2}{7}\right)^2\), each component (both the numerator and the denominator) needs to be squared separately:

• Numerator: \((-2)^2 = 4\)
• Denominator: \(7^2 = 49\)

Then, you combine the squared numerator and denominator to form the new fraction: \(\frac{4}{49}\). Squaring ensures any negative sign in the original fraction is neutralized, leading to a positive result.

Handling Negative Numbers

Negative numbers are less than zero and are usually represented with a minus sign in front, like \(-3\). When you work with negative numbers, especially in multiplication or when dealing with exponents, they have distinct rules:

• A negative number times a positive number results in a negative number.
• A negative number times a negative number results in a positive number. This is why squaring a negative number, resulting in an even number of negative factors, gives a positive result.

In expressions like \(\left(-\frac{2}{7}\right)^2\), although the fraction itself is negative, squaring it results in removing the negative sign because multiplying two negatives yields a positive. So, no matter whether the original fraction is positive or negative, after being squared, it will always be positive. This results in a simplified final result, like \(\frac{4}{49}\), that is free of negative signs.