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Chapter 5: Problem 21

Simplify each expression. Write each result using positive exponents only. See Examples I through 3. $$ 3^{-2}+3^{-1} $$

Short Answer

Expert verified
\( \frac{4}{9} \)

Step by step solution

01

Understanding Negative Exponents

Negative exponents indicate reciprocals. For a base \( a \), \( a^{-n} = \frac{1}{a^n} \). This means that we can rewrite \( 3^{-2} \) as \( \frac{1}{3^2} \) and \( 3^{-1} \) as \( \frac{1}{3^1} \).
02

Rewrite Expressions with Positive Exponents

Apply the rule from Step 1. Rewrite \( 3^{-2} \) as \( \frac{1}{9} \) and \( 3^{-1} \) as \( \frac{1}{3} \). So, the entire expression becomes \( \frac{1}{9} + \frac{1}{3} \).
03

Find a Common Denominator

To add fractions, we need a common denominator. The least common multiple of 9 and 3 is 9. Rewrite \( \frac{1}{3} \) with a denominator of 9, which gives \( \frac{3}{9} \). Thus, our expression becomes \( \frac{1}{9} + \frac{3}{9} \).
04

Add the Fractions

Since both fractions now have the same denominator, simply add the numerators: \( \frac{1}{9} + \frac{3}{9} = \frac{4}{9} \).
05

Simplify the Result

Make sure the result is simplified by checking if the numerator and denominator have common factors. \( \frac{4}{9} \) is already in its simplest form.

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Key Concepts

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

Positive Exponents
Understanding positive exponents is a fundamental aspect of working with expressions in mathematics. An exponent represents how many times a base is multiplied by itself.
  • For example, in the expression \( a^n \), \( n \) is the exponent indicating that the base \( a \) is multiplied by itself \( n \) times.
  • A positive exponent means "multiply." So, \( 3^2 \) equals \( 3 \times 3 = 9 \).
Whenever you work with exponents, you aim to express them in terms of positive exponents, unless specifically instructed otherwise. This process helps simplify expressions and make calculations easier. When simplifying expressions, always ensure that your final answers are expressed using positive exponents to adhere to conventions in mathematics.
Negative Exponents
Negative exponents can be tricky, but they follow a simple rule: they indicate reciprocals rather than direct multiplication.
  • In general, any base \( a \) raised to a negative exponent \( -n \) can be rewritten as \( \frac{1}{a^n} \).
  • For example, \( 3^{-1} \) becomes \( \frac{1}{3} \), and \( 3^{-2} \) becomes \( \frac{1}{9} \).
This rule of flipping the base to its reciprocal and making the exponent positive makes it easier to work with equations and expressions. By converting negative exponents to their positive counterparts, it provides a clearer picture of the value they represent, which is especially useful when adding, subtracting, multiplying, or dividing terms with exponents.
Fraction Addition
Adding fractions is a very useful skill, especially when dealing with algebraic expressions. Unlike whole numbers, fractions must have the same denominator to be added directly.
  • Let's say you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{b} \), with the same denominator.
  • To add these fractions, you simply add the numerators: \( \frac{a}{b} + \frac{c}{b} = \frac{a+c}{b} \).
If the denominators are different, find a common denominator before performing the addition. Properly adding fractions ensures the equation or expression retains its correct value.
Common Denominator
Finding a common denominator is a key step in the adding of fractions. This process makes it possible to add or subtract fractions that do not initially have the same denominator.
  • The common denominator is a multiple that both original denominators can divide into without leaving a remainder.
  • For example, for fractions \( \frac{1}{3} \) and \( \frac{1}{9} \), the common denominator is 9.
  • By rewriting \( \frac{1}{3} \) as \( \frac{3}{9} \), both fractions are now compatible for addition.
Once the fractions have a common denominator, simply add or subtract the numerators while retaining the denominator. This ensures accurate results when working with fractional expressions.

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Most popular questions from this chapter

Multiply each expression. $$ -4 a\left(3 a^{2}-4\right) $$

Match each expression on the left to the equivalent expression on the right. See the Concept Check in this section. $$ (a-b)(a+b) $$ $$A. a^{2}-b^{2}$$ $$B. a^{2}+b^{2}$$ $$C. a^{2}-2 a b+b^{2}$$ $$D. a^{2}+2 a b+b^{2}$$ $$E. none of these$$

Multiply. $$(5 x+4)\left(x^{2}-x+4\right)$$

Perform the indicated operation. $$\left(-7 m^{2}\right)^{2}$$

Explain how to check a polynomial long division result when the remainder is not \(0 .\)
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