Problem 12 - Beginning and Intermediate Algebra

Question

Evaluate. \(\left(\frac{1}{2}\right)^{5}\)

Step-by-Step Solution

Answer

The result of \( \left( \frac{1}{2} \right)^{5} \) is \( \frac{1}{32} \).

Step 1: Understand the Expression

The expression \( \left( \frac{1}{2} \right)^{5} \) means that the fraction \( \frac{1}{2} \) is multiplied by itself 5 times. This is called raising a number to a power.

Step 2: Calculate Step-by-Step Multiplication

Start by multiplying \( \frac{1}{2} \) by itself:1. Multiply once: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)2. Multiply again: \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \)3. Continue: \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \)4. And again: \( \frac{1}{16} \times \frac{1}{2} = \frac{1}{32} \)

Step 3: Finalize the Evaluation

After multiplying \( \frac{1}{2} \) by itself five times, we get the product \( \frac{1}{32} \). This is the value of \( \left( \frac{1}{2} \right)^{5} \).

Key Concepts

Fraction Multiplication

Fraction multiplication is a key concept in mathematics that involves multiplying two fractions to get a new value. In this context, it’s essential to understand that when you multiply fractions, you multiply the numerators together and the denominators together.
For example, if you are multiplying two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), the result will be \( \frac{a \times c}{b \times d} \). There's no need to find a common denominator as you do with fraction addition or subtraction, making it simpler in one way.
However, when it comes to raising a fraction to a power, the multiplication is done repeatedly with the same fraction. This means that if you are asked to evaluate \( \left(\frac{1}{2}\right)^5\), you would work with the fraction \( \frac{1}{2} \) and multiply it by itself five times, which brings us to another important concept: evaluating powers.

Evaluating Powers

Evaluating powers involves understanding how to express a number that is multiplied by itself a number of times. In general, if you have a number \( x \) and you are looking to evaluate \( x^n \), it means you multiply \( x \) by itself \( n \) times. So \( x^1 \) is simply \( x \), \( x^2 \) is \( x \times x \), \( x^3 \) is \( x \times x \times x \), and so on.
When evaluating a fraction raised to a power, like \( \left( \frac{1}{2} \right)^5 \), you apply the same principle. The fraction \( \frac{1}{2} \) is multiplied by itself five times. You could express this as:

• First multiplication: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
• Second multiplication: \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \)
• Third multiplication: \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \)
• Continue: \( \frac{1}{16} \times \frac{1}{2} = \frac{1}{32} \)

This process demonstrates how each stage builds on the previous result, reducing the fraction at each step until reaching the final value, \( \frac{1}{32} \). This step-by-step approach solidifies the understanding of multiplication and power evaluation in fractions.

Step-by-Step Solution

Breaking down problems into step-by-step solutions helps clarify complex mathematical concepts. This systematic approach involves understanding each stage before moving to the next, ensuring no step is skipped. Take the problem \( \left( \frac{1}{2} \right)^5 \) as an example.
**Step 1**: Understand the Expression - Grasp what \( \left( \frac{1}{2} \right)^5 \) means. Recognize it as \( \frac{1}{2} \) being multiplied by itself 5 times.
**Step 2**: Perform the Multiplications - Multiply \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \), then \( \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \), followed by \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \), and last, \( \frac{1}{16} \times \frac{1}{2} = \frac{1}{32} \). Each step guides you closer to the final answer, reinforcing the methodical procedure.
**Step 3**: Conclude with the Final Result - Conclude by affirming that after 5 repeated multiplications, you arrive at \( \frac{1}{32} \).
This structure is not only helpful for evaluating powers but can also be applied broadly in math problems to reveal each step's purpose and ensure comprehension and accuracy.