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Chapter 0: Problem 56

Use the properties of exponents to simplify each expression. (See Examples \(3-6)\) $$ \left(-\frac{1}{3}-\frac{1}{4}+\frac{1}{2}\right)^{-2} $$

Short Answer

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Step by step solution

01

Simplify the Expression Inside the Parentheses

First, combine the fractions inside the parentheses. Start by finding a common denominator for the fractions \(-\frac{1}{3}, -\frac{1}{4}, \text{and} \frac{1}{2}\). The common denominator for 3, 4, and 2 is 12.Rewrite each fraction with the common denominator:\[-\frac{1}{3} = -\frac{4}{12}\]\[-\frac{1}{4} = -\frac{3}{12}\]\[\frac{1}{2} = \frac{6}{12}\]Combine these results:\[-\frac{4}{12} - \frac{3}{12} + \frac{6}{12} = -\frac{4}{12} - \frac{3}{12}+ \frac{6}{12}= -\frac{1}{12}\]
02

Simplify the Parentheses

With the combined result of the fractions, the expression now looks like this:\[(-\frac{1}{12})^{-2}\]
03

Apply the Negative Exponent Rule

Recall that for any expression \(a^{-n} = \frac{1}{a^n}\). Use this rule to simplify \((-\frac{1}{12})^{-2}\):\[\left(-\frac{1}{12}\right)^{-2} = \frac{1}{(-\frac{1}{12})^2}\]
04

Simplify the denominator

Compute the square of the denominator:\[\left(-\frac{1}{12}\right)^2 = \frac{1}{144}\]Thus, the expression becomes \[\frac{1}{\frac{1}{144}} = 144\]

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Key Concepts

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

Simplifying Fractions
To simplify expressions involving fractions, we need to combine them into a single fraction. Here, you'll often start by finding a common denominator. The common denominator is a shared multiple of the denominators of the fractions. For example, in the given exercise, the fractions \(-\frac{1}{3}\), \(-\frac{1}{4}\), and \(\frac{1}{2}\) were combined by using 12 as the common denominator. This process allows us to rewrite each fraction so they can be easily added or subtracted.

To recap, you should:
  • Find the least common denominator of the fractions involved
  • Convert each fraction to an equivalent fraction with this common denominator
  • Add or subtract the numerators, keeping the common denominator

This results in a single simplified fraction, which is essential for further operations.
Negative Exponents
Negative exponents might seem tricky, but they're simpler than they appear. A negative exponent indicates that you should take the reciprocal of the base and then apply the positive exponent. For instance, \(a^{-n} = \frac{1}{a^n}\).

In the provided solution, the expression \((- \frac{1}{12})^{-2}\) transforms into \(\frac{1}{(-\frac{1}{12})^2}\) by applying this rule. This step is crucial because it breaks down the problem into a more manageable form, essentially turning a negative exponent into a positive one by flipping the base.

Always remember:
  • A negative exponent means 'reciprocal'.
  • Convert the base to its reciprocal and then perform the exponentiation as usual
Common Denominator
A key part of working with fractions is finding a common denominator. This is particularly useful when adding or subtracting fractions, as it ensures all fractions are on a level playing field.

In our example, the common denominator for 3, 4, and 2 is 12. Each fraction was then rewritten with 12 as the denominator:
  • \(- \frac{1}{3} = - \frac{4}{12}\)
  • \(- \frac{1}{4} = - \frac{3}{12}\)
  • \( \frac{1}{2} = \frac{6}{12}\)

This step is critical because it allows the fractions to be easily combined.

A few steps to find a common denominator:
  • Identify the least common multiple (LCM) of the denominators
  • Rewrite each fraction to have this common denominator by multiplying the numerator and denominator by the same number
  • Proceed with the addition or subtraction of the fractions

Mastering common denominators will unlock many more advanced fractional operations.

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Most popular questions from this chapter

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