/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 59

Evaluate each expression. $$ 5^{-1}+6^{-1} $$

Short Answer

Expert verified
\(\frac{11}{30}\)

Step by step solution

01

Evaluate the Reciprocal of 5

The expression contains the term \(5^{-1}\). To find the reciprocal of 5, calculate \(5^{-1} = \frac{1}{5}\).
02

Evaluate the Reciprocal of 6

Similarly, the expression contains the term \(6^{-1}\). To find the reciprocal of 6, calculate \(6^{-1} = \frac{1}{6}\).
03

Add the Reciprocals

Now add the two reciprocals: \(\frac{1}{5} + \frac{1}{6}\). To add these fractions, find a common denominator. The least common multiple of 5 and 6 is 30. Rewrite each fraction with this common denominator: \(\frac{1}{5} = \frac{6}{30}\) and \(\frac{1}{6} = \frac{5}{30}\). Now add them together: \(\frac{6}{30} + \frac{5}{30} = \frac{11}{30}\).

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Key Concepts

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

evaluating expressions
Evaluating expressions means finding the value of a mathematical expression. In this context, the expression involves negative exponents. When you see something like \(a^{-1}\), it means you need to find the reciprocal, or the multiplicative inverse, of \(a\). For example, \(5^{-1}\) equals \(\frac{1}{5}\). This is because any number raised to the power of -1 is equal to 1 divided by that number. Understanding this concept helps when you simplify or solve expressions involving negative exponents.
fractions addition
Adding fractions involves a couple of steps. First, make sure the fractions have a common denominator. If they don鈥檛, you have to find one. Once both fractions have the same denominator, you can add the numerators together and keep the denominator the same. Taking our example, after finding the reciprocals of 5 and 6, we need to add \(\frac{1}{5}\) and \(\frac{1}{6}\). We first find a common denominator, which allows us to rewrite these fractions and then add them. This process turns two fractions into one that is easier to work with.
common denominator
A common denominator is a shared multiple of the denominators of two or more fractions. It's crucial when adding or subtracting fractions because it allows you to combine the fractions into a single fraction. For our example with \(\frac{1}{5}\) and \(\frac{1}{6}\), the least common denominator is 30. We convert \(\frac{1}{5}\) and \(\frac{1}{6}\) into \(\frac{6}{30}\) and \(\frac{5}{30}\) respectively. Now you can easily add them: \(\frac{6}{30} + \frac{5}{30} = \frac{11}{30}\). This makes the task of adding fractions straightforward and manageable.

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