Problem 83 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ \frac{1}{3}\left(\frac{1}{6}\right)-\left(-\frac{1}{3}\right)^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{1}{18}\).

Step 1: Solve the Multiplication

Begin by solving the multiplication inside the expression: \( \frac{1}{3} \times \frac{1}{6} \). To multiply fractions, multiply the numerators and multiply the denominators.\[\frac{1 \times 1}{3 \times 6} = \frac{1}{18}\]

Step 2: Solve the Exponentiation

Next, solve the exponentiation in the expression \( \left(-\frac{1}{3}\right)^{2} \). Squaring a fraction involves squaring the numerator and the denominator. Since it's a negative, the square will be positive:\[\left(-\frac{1}{3}\right)^{2} = \frac{(-1)^{2}}{3^{2}} = \frac{1}{9}\]

Step 3: Substitute and Simplify

Substitute the results from Steps 1 and 2 back into the original expression. This gives us:\[ \frac{1}{18} - \frac{1}{9}\]Convert \( \frac{1}{9} \) to a common denominator of 18 (since \( \frac{1}{9} = \frac{2}{18} \)) and perform the subtraction:\[\frac{1}{18} - \frac{2}{18} = \frac{1-2}{18} = -\frac{1}{18}\]

Step 4: Simplify the Expression Fully

The expression does not reduce further, so the simplified result of the expression is \(-\frac{1}{18}\).

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward process that involves only a few simple steps. When we multiply fractions, we need to multiply the numerators together and the denominators together. This allows us to find a new fraction that is equivalent to the product of the original fractions. Let's take a closer look with an example provided in the problem:

• Consider the multiplication: \(\frac{1}{3} \times \frac{1}{6}\).

Multiply the numerators (the top numbers of the fractions): \(1 \times 1 = 1\). Then multiply the denominators (the bottom numbers of the fractions): \(3 \times 6 = 18\).
So, \(\frac{1}{3} \times \frac{1}{6} = \frac{1}{18}\). This simple rule helps you multiply fractions quickly and precisely. Always simplify your final answer if possible, though in this case, \(\frac{1}{18}\) is already in its simplest form.

Fraction Exponentiation

Fraction exponentiation may seem tricky at first, but it's actually quite simple once you get the hang of it. When you raise a fraction to a power, you must raise both the numerator and the denominator to this power separately.

Let's take the example from the exercise: \(\left(-\frac{1}{3}\right)^{2}\).

• The base here is \(-\frac{1}{3}\), and the exponent is 2.

Squaring a fraction means multiplying the fraction by itself. Since the exponentiation of a negative number results in a positive number when the exponent is even, we have:\[(-1)^2 = 1 \quad \text{and} \quad 3^2 = 9\]Therefore, \(\left(-\frac{1}{3}\right)^{2} = \frac{1}{9}\). Remember, if the exponent were odd, the result would be negative.

Subtraction of Fractions

Subtracting fractions requires a common denominator. This ensures that the fractions are being compared or subtracted on the same scale. Here's how it's done using the expression from the textbook solution:

• We need to subtract \(\frac{1}{9}\) from \(\frac{1}{18}\).

The first step is to convert \(\frac{1}{9}\) to an equivalent fraction with a denominator of 18. By multiplying both the numerator and the denominator by 2, we get \(\frac{2}{18}\). Now, subtract the two fractions:\[\frac{1}{18} - \frac{2}{18} = \frac{1-2}{18} = -\frac{1}{18}\]Always simplify your answer when possible. In this case, \(-\frac{1}{18}\) is the simplest form. Using a common denominator is key to effectively subtracting any fractions.