Question

$$ \left(-\frac{1}{3}\right)^2 $$ A. \(-\frac{2}{9}\) B. \(-\frac{1}{9}\) C. \(\frac{1}{9}\) D. \(\frac{2}{9}\)

Short answer

The correct answer is C. \(\frac{1}{9}\).

Step-by-step solution

Understand the Problem

Calculate the square of \(-\frac{1}{3}\). The square of a fraction involves squaring both the numerator and the denominator.

Square the Numerator

Square the numerator of the fraction. The numerator is \(-1\). Squaring \(-1\) gives \((-1)^2 = 1\).

Square the Denominator

Square the denominator of the fraction. The denominator is \(3\). Squaring \(3\) gives \((3)^2 = 9\).

Combine the Results

The square of the fraction \(-\frac{1}{3}\) is \frac{1}{9}\. Therefore, the result is \frac{1}{9}\.

Select the Correct Answer

From the given options, the correct answer is \(\frac{1}{9}\), which corresponds to option C.

Key concepts

headline of the respective core concept

To understand the squaring of fractions, it's important to know what fractions are.

headline of the respective core concept

When dealing with exponents in fractions, it increases the comprehension of applied operations.
The primary task is to square the fraction \(-\frac{1}{3}\). An exponent expresses the quantity of times an element needs to be multiplied by itself. In this instance, we are working with the exponent 2.
To solve the fraction, apply the squaring technique to both the numerator and the denominator:
1. The numerator of the fraction is -1. Squaring -1 gives \((-1)^2 = 1\).
2. The denominator of the fraction is 3. Squaring 3 gives \(3^2 = 9\).
Combining them, the final result of \(-\frac{1}{3}\) yields \(\frac{1}{9}\). Achieving the correct answer as provided earlier.
Mastering this concept makes it easy to handle any fraction with an exponent.

headline of the respective core concept

The elements of fractions are key to understanding why squaring works as shown.
Fractions are mathematical expressions representing the division of one quantity by another.
Each fraction consists of two major components:

• The Numerator - This is the top number in a fraction representing the part.
• The Denominator - This is the bottom number in a fraction representing the total.

For the fraction \(-\frac{1}{3}\), the numerator is -1, and the denominator is 3.
When squaring \(-\frac{1}{3}\), square both elements separately:
1. The Numerator: Squaring -1 results in +1 as \((-1)^2 = 1\).
2. The Denominator: Squaring 3 results in 9 as \(3^2 = 9\).
Finally, combining the squared elements provides the fraction \(\frac{1}{9}\), making understanding and solving fractional problems much simpler.