Problem 21 - Fundamentals of Mathematics

Question

Find the value of each of the following. $$\left(\frac{1}{8}\right)^{2}$$

Step-by-Step Solution

Answer

\( \frac{1}{64} \)

Step 1: Identify the Base and Exponent

The expression given is \( \left(\frac{1}{8}\right)^2 \). Here, the base is \( \frac{1}{8} \) and the exponent is 2. This means we need to multiply \( \frac{1}{8} \) by itself.

Step 2: Multiply the Base by Itself

Since the exponent is 2, we multiply the base, \( \frac{1}{8} \), by itself: \[ \frac{1}{8} \times \frac{1}{8} \]

Step 3: Perform the Multiplication

To multiply fractions, multiply the numerators together and the denominators together: \[ \frac{1 \times 1}{8 \times 8} = \frac{1}{64} \]

Step 4: Simplify the Result

The fraction \( \frac{1}{64} \) is already in its simplest form because 1 is the smallest positive integer and 64 has no common factors with 1 other than 1 itself.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. A fraction consists of two numbers: a numerator and a denominator. The numerator is the top number and indicates how many parts of the whole you have. The denominator is the bottom number and shows into how many parts that whole is divided. For example, in \( \frac{1}{8} \), the numerator is 1, meaning we have one part out of a total of 8 parts, which is the denominator. This fraction therefore represents one eighth of a whole.
Understanding fractions is crucial in arithmetic, whether you're splitting a pizza into slices or calculating probabilities. Fractions can express numbers smaller than 1 or larger than 1, depending on whether the numerator is less than or greater than the denominator.

How to Multiply Fractions

Multiplying fractions might seem tricky at first, but it's quite straightforward once you get the hang of it. To multiply two fractions, you multiply the numerators to get the new numerator and the denominators to get the new denominator. For example, if you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), their product will be \( \frac{a \times c}{b \times d} \).

• Multiply the numerators of the fractions.
• Multiply the denominators of the fractions.
• The result is the product of these two fractions.

Let's apply this to our example \( \frac{1}{8} \times \frac{1}{8} \). You multiply the numerators: 1 \( \times \) 1 = 1. Then, you multiply the denominators: 8 \( \times \) 8 = 64. So, the result of the multiplication is \( \frac{1}{64} \). It's simple after a bit of practice!

The Process of Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form where the numerator and the denominator have no common factors other than 1. This process makes the fraction easier to work with and understand.
To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The resulting fraction is the simplified one.

In the case of \( \frac{1}{64} \), the numerator 1 and the denominator 64 have no other common factor except for 1, which means this fraction is already in its simplest form. Simplifying is essential in various calculations for clarity and precision. It comes in handy when adding, subtracting, or comparing fractions as well.