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Chapter 11: Problem 39

Express each of the given expressions in simplest form with only positive exponents. $$2 \times 3^{-1}+4 \times 3^{-2}$$

Short Answer

Expert verified
The expression simplifies to \(\frac{10}{9}\).

Step by step solution

01

Simplify Individual Terms

The given expression is \(2 \times 3^{-1} + 4 \times 3^{-2}\). Start by simplifying each term separately. The first term \(2 \times 3^{-1}\) can be rewritten as \(\frac{2}{3}\) because \(3^{-1} = \frac{1}{3}\). The second term \(4 \times 3^{-2}\) can be rewritten as \(\frac{4}{9}\) because \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\).
02

Combine Terms with a Common Denominator

To combine \(\frac{2}{3}\) and \(\frac{4}{9}\), find a common denominator. The least common denominator of 3 and 9 is 9. Rewrite \(\frac{2}{3}\) with a denominator of 9: \(\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}\).
03

Add the Fractions

Now that both fractions have the same denominator, add them: \(\frac{6}{9} + \frac{4}{9} = \frac{10}{9}\). This is the simplified form of the expression.

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Key Concepts

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

Positive Exponents
In algebra, understanding exponents is crucial. Exponents indicate how many times a number, the base, is multiplied by itself. A positive exponent signifies standard multiplication.
For example, if we have a term like \(3^2\), it means that you multiply 3 by itself, resulting in \(3 \times 3 = 9\). Hence, \(3^2 = 9\).
However, sometimes we encounter negative exponents, like \(3^{-1}\). This is where it gets interesting鈥攏egative exponents are a way to express division rather than multiplication. Specifically, \(a^{-n} = \frac{1}{a^n}\). Thus, \(3^{-1} = \frac{1}{3}\).
  • Positive exponents relate to repeated multiplication.
  • Negative exponents indicate division, flipping the base.
  • Converting negative to positive exponents helps simplify expressions.
Fractions
Fractions are a central concept in algebra, representing parts of a whole. They appear frequently when simplifying expressions and equations. A fraction is composed of a numerator (top number) and a denominator (bottom number).
For instance, the fraction \(\frac{3}{4}\) means 3 parts out of 4 equal parts. It鈥檚 crucial to understand how fractions operate to combine or simplify them efficiently.
  • Add or subtract fractions by finding a common denominator.
  • Multiplication of fractions is straightforward鈥攎ultiply numerators and denominators. \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\).
  • Division of fractions requires flipping the second fraction and multiplying.
Getting comfortable with fractions enables you to manipulate expressions effectively.
Common Denominator
A key step in adding fractions is to have a common denominator. This allows the numerators to be directly added or subtracted. The common denominator must be a shared multiple of the original denominators.
Here are the steps to find a common denominator:
  • Identify the denominators of each fraction.
  • Calculate the least common multiple (LCM) of these denominators.
  • Rewrite each fraction with the common denominator by scaling the numerators accordingly.
For example, with \(\frac{2}{3}\) and \(\frac{4}{9}\), the common denominator is 9. Change \(\frac{2}{3}\) to \(\frac{6}{9}\) to match \(\frac{4}{9}\). Now they can be easily added as \(\frac{6}{9} + \frac{4}{9} = \frac{10}{9}\). Understanding common denominators simplifies operations with fractions and is an essential skill in algebra.

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Most popular questions from this chapter

Perform the indicated operations. (a) Express the radius of a sphere as a function of its volume \(V\) using fractional exponents. (b) If the volume of the moon is \(2.19 \times 10^{19} \mathrm{m}^{3},\) find its radius.

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$y^{9} y^{-2}$$

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$(-4)^{0}$$

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$\left(\frac{3 a^{2}}{4 b}\right)^{-3}\left(\frac{4}{a}\right)^{-5}$$

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$\left(R_{1}^{-1}+R_{2}^{-1}\right)^{-1}$$
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