Problem 84 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ \frac{1}{2}\left(\frac{1}{8}\right)+\left(-\frac{1}{4}\right)^{2} $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{1}{8} \).

Step 1: Simplify the Fraction Multiplication

The expression given is \( \frac{1}{2}\left(\frac{1}{8}\right) + \left(-\frac{1}{4}\right)^{2} \). First, we need to multiply \( \frac{1}{2} \) by \( \frac{1}{8} \). When multiplying fractions, multiply the numerators and the denominators: \( \frac{1 \times 1}{2 \times 8} = \frac{1}{16} \).

Step 2: Solve the Squaring of a Fraction

The next part of the expression is \( \left(-\frac{1}{4}\right)^{2} \). When a fraction is squared, both the numerator and the denominator are squared: \( \left(-\frac{1}{4}\right)^{2} = \frac{(-1)^{2}}{(4)^{2}} = \frac{1}{16} \).

Step 3: Add the Results

Now, add the two results: \( \frac{1}{16} + \frac{1}{16} \). Since the denominators are the same, we can directly add the numerators: \( \frac{1 + 1}{16} = \frac{2}{16} \). Simplify \( \frac{2}{16} \) by dividing the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{2}{16} = \frac{1}{8} \).

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward process. All you need to do is multiply the numerators together and then the denominators together.
This means, if you are looking at a fraction multiplication like \( \frac{a}{b} \times \frac{c}{d} \), you would compute this as \( \frac{a \times c}{b \times d} \). For instance, in our example, we multiplied \( \frac{1}{2} \) and \( \frac{1}{8} \), resulting in \( \frac{1 \times 1}{2 \times 8} = \frac{1}{16} \).
**Key Points:**

• Always multiply straight across: top with top (numerators) and bottom with bottom (denominators).
• If possible, simplify the result by finding the greatest common divisor for both the numerator and denominator before and after multiplication.
• If the fractions contain mixed numbers, convert them into improper fractions first before multiplying.

Squaring Fractions

Squaring a fraction involves raising both the numerator and the denominator to the power of two.
This means that given a fraction \( \frac{a}{b} \), squaring it will give you \( \frac{a^2}{b^2} \).
In the example, we had to square \( \left(-\frac{1}{4}\right) \). This resulted in \( \frac{(-1)^2}{4^2} \), which simplifies to \( \frac{1}{16} \) because \((-1)^2 = 1\) and \((4)^2 = 16\).
**Things to Remember:**

• The square of any negative number will always be positive, due to the property \((-x)^2 = x^2\).
• Ensure that entire fractions are enclosed in parentheses before squaring, to avoid mistakes.
• Double-check your math, as errors in squaring numbers are common.

Adding Fractions

Adding fractions requires the fractions to have a common denominator.
If the denominators are the same, you can directly add the numerators.
When you have fractions like \( \frac{m}{n} + \frac{p}{n} \), the sum is \( \frac{m+p}{n} \). In our example, we added \( \frac{1}{16} + \frac{1}{16} \) because they share the same denominator.
This calculation led to \( \frac{1+1}{16} = \frac{2}{16} \), which simplifies to \( \frac{1}{8} \).
**Helpful Tips:**

• If the denominators are different, find a common denominator before adding.
• Simplify fractions as much as possible for clearer results.
• When using a common denominator, it should be the least common multiple for efficiency.