Problem 105 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{8} \).

Step 1: Distributing the Multiplication

Evaluate \( \frac{1}{2} \times \frac{1}{8} \) as part of the expression. Multiplying the two fractions gives: \( \frac{1}{2} \times \frac{1}{8} = \frac{1}{16} \). Thus, the expression simplifies to: \( \frac{1}{16} + \left(-\frac{1}{4}\right)^2 \).

Step 2: Evaluating the Square

Evaluate \( \left(-\frac{1}{4} \right)^2 \). Squaring a fraction involves squaring both the numerator and denominator. Therefore, \( \left(-\frac{1}{4} \right)^2 = \frac{1}{16} \). Now the expression is: \( \frac{1}{16} + \frac{1}{16} \).

Step 3: Adding the Fractions

Add the fractions: \( \frac{1}{16} + \frac{1}{16} = \frac{2}{16} \). Simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 2, to obtain \( \frac{1}{8} \).

Step 4: Final Result

The simplified result of the expression \( \frac{1}{2}\left(\frac{1}{8}\right)+\left(-\frac{1}{4}\right)^{2} \) is \( \frac{1}{8} \).

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward process once you understand the basics. When you multiply two fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For the given exercise, we start by multiplying \( \frac{1}{2} \) by \( \frac{1}{8} \). Here's how it works:

• Numerators: \(1 \times 1 = 1\)
• Denominators: \(2 \times 8 = 16\)

Thus, \( \frac{1}{2} \times \frac{1}{8} = \frac{1}{16} \).
Multiplication is all about making the calculation as simple as possible. The order in which you multiply fractions does not matter—multiplication of fractions is commutative, which means you can multiply in any order and still get the same result.

Squaring Fractions

Squaring a fraction involves multiplying the fraction by itself. It's important to remember that squaring affects both the numerator and the denominator. For instance, when squaring \(-\frac{1}{4}\), you perform the operation on both the top and bottom of the fraction:

• Square the numerator: \((-1)^2 = 1\)
• Square the denominator: \(4^2 = 16\)

The result of squaring \(-\frac{1}{4}\) is \(\frac{1}{16}\).
Notice that even though we started with a negative fraction, squaring it makes the result positive. This is because a negative times a negative is always positive. It's a good rule of thumb to remember when dealing with squares.

Adding Fractions

When adding fractions, it's crucial to ensure that the denominators are the same. If they aren't, you'll need to find a common denominator before you can add them. In this exercise, the fractions \( \frac{1}{16} \) and \( \frac{1}{16} \) already share the same denominator, which makes the process easier:

• Add the numerators: \(1 + 1 = 2\)
• Keep the denominator the same: 16

This gives us \( \frac{2}{16} \), which can be simplified. You simplify a fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of 2 and 16 is 2, so:

• Numerator: \(2 \div 2 = 1\)
• Denominator: \(16 \div 2 = 8\)

Hence, \( \frac{2}{16} \) simplifies to \( \frac{1}{8} \).
Adding fractions is all about making calculations simpler by using common rules and simplifications.