/*! 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 60 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 60

Evaluate each expression. $$ 2^{-1}+8^{-1} $$

Short Answer

Expert verified
\(\frac{5}{8}\)

Step by step solution

01

Understand the negative exponent

A negative exponent means taking the reciprocal of the base. For example, if the term is \(a^{-1}\), it is equivalent to \(\frac{1}{a}\). Apply this to each term.
02

Convert each term

Convert each term using the reciprocal rule: the first term: \(2^{-1} = \frac{1}{2}\)the second term: \(8^{-1} = \frac{1}{8}\)
03

Add the fractions

To add the fractions \(\frac{1}{2}\) and \(\frac{1}{8}\), find a common denominator. The least common denominator of 2 and 8 is 8:\(\frac{1}{2} = \frac{4}{8}\)Now add the two fractions: \(\frac{4}{8} + \frac{1}{8} = \frac{4+1}{8} = \frac{5}{8}\)

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Key Concepts

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

Reciprocal of a number
The reciprocal of a number is simply one divided by that number. For instance, the reciprocal of 5 is \(\frac{1}{5}\). Reciprocals are helpful when dealing with negative exponents. For example, \(\text{a}^{-1}\) becomes \(\frac{1}{\text{a}}\). Here's a tip: swapping the numerator and the denominator also gives a reciprocal. So, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
If you see an expression like \(\text{b}^{-3}\), first find the reciprocal: \(\frac{1}{\text{b}}\). Then, raise it to the power of 3, resulting in \(\frac{1}{\text{b}^3}\).
This trick makes it easier to handle negative exponents in any calculation!
Least Common Denominator
When adding fractions, the least common denominator (LCD) is the smallest number that can evenly divide both denominators. This is crucial for making the addition simpler. For instance, to add \(\frac{1}{2}\) and \(\frac{1}{8}\), you need the LCD of 2 and 8. Here, the LCD is 8.
Why 8? Because 8 is the smallest number that both 2 and 8 can divide into without leaving a remainder. To use the LCD:
  • Convert \(\frac{1}{2}\) to \(\frac{4}{8}\) since \(\frac{1}{2} equals \frac{4}{8}\)
  • Now, you have common denominators: \(\frac{4}{8}\) and \(\frac{1}{8}\)
This step is very important before proceeding to add fractions easily!
Adding Fractions
Adding fractions is straightforward if they have the same denominator. Simply add the numerators and keep the denominator unchanged. For example, \(\frac{4}{8} + \frac{1}{8} = \frac{4+1}{8}\), which simplifies to \(\frac{5}{8}\).
However, if fractions don't have the same denominator, find the least common denominator (LCD) first and convert the fractions. Here's the detailed addition process:
  • Find the LCD of the denominators.
  • Adjust each fraction to have the LCD as the new denominator.
  • Add the numerators of these equivalent fractions.
With practice, adding fractions becomes a simple task. Always check for a common denominator first, and you鈥檒l be set!

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Most popular questions from this chapter

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{1}{5 x}\right)^{-3} $$

Perform the indicated operations. $$ \left[-\left(y^{4}-y^{2}+1\right)-\left(y^{4}+2 y^{2}+1\right)\right]+\left(3 y^{4}-3 y^{2}-2\right) $$

Find each product. $$ [(m+p)-5][(m+p)+5] $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ -2 m^{-1}\left(m^{3}\right)^{2} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{(-8 x y) y^{3}}{4 x^{5} y^{4}} $$
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