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Chapter 6: Problem 171

Jorge thinks \(\left(\frac{1}{2}\right)^{2}\) is \(1 .\) What is wrong with his reasoning?

Short Answer

Expert verified
Jorge's mistake is that \(\bigg(\frac{1}{2}\bigg)^2 = \frac{1}{4}\) not 1.

Step by step solution

01

Understanding Exponents

Review the definition of an exponent. When you raise a fraction to a power, you raise both the numerator and the denominator to that power. Specifically, \(\frac{a}{b}\big)^n = \frac{a^n}{b^n}\).
02

Apply the Exponent Rule

Apply this rule to the given fraction: \(\bigg(\frac{1}{2}\bigg)^2\) becomes \(\frac{1^2}{2^2}\). Compute the results for the numerator and the denominator.
03

Simplify

Simplify the result. \(1^2 = 1\) and \(2^2 = 4\). Therefore, \(\bigg(\frac{1}{2}\bigg)^2 = \frac{1}{4}\).
04

Identify the Error

Identify where the mistake was made. Jorge mistakenly thought that \(\bigg(\frac{1}{2}\bigg)^2\) equaled 1, but actually \(\frac{1}{4}\) is the correct answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Exponents
Fraction exponents, also known as fractional exponents, are where the exponent is a fraction. They can be tricky at first but are quite simple to work with once you understand them. For example, \(\bigg(\frac{a}{b}\bigg)^n\) means you raise both the numerator (a) and the denominator (b) to the power of n. So \(\bigg(\frac{1}{2}\bigg)^2\) results in \(\frac{1^2}{2^2}\). This means you must compute \(\frac{1}{4}\), not 1. Remember, fractions are just divisions, and these rules apply just like they do with whole numbers.
Simplifying Exponents
Simplifying exponents involves applying the exponent rules properly to make equations easier. Here's how you simplify fraction exponents step-by-step:
  • Identify the base numbers and the exponents.
  • Apply the exponent to both the numerator and denominator: \(\bigg(\frac{a}{b}\bigg)^n\) becomes \(\frac{a^n}{b^n}\).
  • Compute the power of the numerator and the power of the denominator.
  • Simplify the fraction if possible.
For instance, \(\bigg(\frac{1}{2}\bigg)^2\) becomes \(\frac{1^2}{2^2}\). This simplifies to \(\frac{1}{4}\). This method ensures that you correctly handle the exponent and simplify the calculation accurately.
Error Correction in Exponents
Identifying errors when working with exponents is crucial, especially in fractions. In Jorge鈥檚 case, he thought \(\bigg(\frac{1}{2}\bigg)^2\) equals 1. Here鈥檚 the mistake:
  • First, recognize that the exponent applies to both the numerator and denominator.
  • Second, compute each part separately: \(\bigg(\frac{1}{2}\bigg)^2\) means \(\frac{1^2}{2^2}\).
  • Finally, simplify the results: \(\frac{1}{4}\).
Remember, an exponent on a fraction means you multiply the fraction by itself for the specified number of times. Error correction is about understanding and catching these mistakes early to ensure you get to the correct answer.

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