Problem 77 - Prealgebra

Question

Use the definition of exponents to simplify each expression. \(\left(\frac{1}{2}\right)^{3}\)

Step-by-Step Solution

Answer

\(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\).

Step 1: Understanding the Base and Exponent

Identify the base and the exponent. In this expression, the base is \(\frac{1}{2}\) and the exponent is 3. The exponent tells us how many times to multiply the base by itself.

Step 2: Apply the Definition of Exponents

According to the definition of exponents, \(a^n\) means to multiply \(a\) by itself \(n\) times. Thus, \(\left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\).

Step 3: Perform the Multiplication

Multiply the fractions step by step: First, calculate \(\frac{1}{2} \times \frac{1}{2} = \frac{1\times1}{2\times2} = \frac{1}{4}\). Next, multiply by \(\frac{1}{2}\) again: \(\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}\).

Step 4: Conclusion: Verify the Result

By performing the multiplication carefully, we find that \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\). Double-check each multiplication to ensure accuracy.

Key Concepts

Base and Exponent

In mathematical expressions involving exponents, there are two main components: the base and the exponent. Understanding these is essential to working with exponents effectively.

The **base** is the number that is being multiplied. In the expression \(\left(\frac{1}{2}\right)^{3}\), the base is \(\frac{1}{2}\). The **exponent** tells us how many times the base is multiplied by itself. Here, the exponent is 3, meaning \(\frac{1}{2}\) is multiplied by itself three times.

To visualize it, think of the base as a number that you place a bet on and the exponent as the number of times you double down on that bet. If the base is your starting point, the exponent is the powerhouse pushing you forward with repeated multiplication.

Multiplying Fractions

Multiplying fractions might seem tricky at first, but with a few simple steps, it becomes straightforward.

When you multiply fractions, you simply multiply the numerators (the top numbers) together and then the denominators (the bottom numbers) together.

• Example: To multiply \(\frac{1}{2} \times \frac{1}{2}\), multiply the numerators: \(1 \times 1 = 1\), and the denominators: \(2 \times 2 = 4\). The result is \(\frac{1}{4}\).
• Then, to multiply \(\frac{1}{4} \times \frac{1}{2}\), you follow the same method. Multiply the numerators: \(1 \times 1 = 1\), and the denominators: \(4 \times 2 = 8\). The result is \(\frac{1}{8}\).

By methodically following these steps, you can confidently multiply any set of fractions.

Definition of Exponents

Exponents give a concise way to express repeated multiplication of the same number.

The definition of exponents is represented as \(a^n\), where \(a\) is the base and \(n\) is the exponent.

This notation signifies multiplying \(a\) by itself for a total of \(n\) times. For instance, in \(\left(\frac{1}{2}\right)^3\), the expression can be expanded to \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\).

Exponents offer a simple way to handle large numbers without writing them out in full. This idea is crucial in mathematics, as it keeps expressions neat and manageable. The key takeaway: the exponent acts as a shorthand notation for repeated multiplication.