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Chapter 0: Problem 22

Evaluate each expression. $$ 3^{-1}-3^{-3} $$

Short Answer

Expert verified
The expression evaluates to \(\frac{8}{27}\).

Step by step solution

01

Understand Negative Exponents

Recall that a negative exponent means the reciprocal of the base raised to the corresponding positive exponent. For example, \[a^{-n} = \frac{1}{a^n}\].
02

Evaluate 3^{-1}

Using the rule for negative exponents, \[3^{-1} = \frac{1}{3^1} = \frac{1}{3}\].
03

Evaluate 3^{-3}

Following the same rule for negative exponents, \[3^{-3} = \frac{1}{3^3} = \frac{1}{27}\].
04

Subtract the Values

Now subtract the two fractions obtained earlier: \[\frac{1}{3} - \frac{1}{27}\].To subtract these fractions, find a common denominator. The least common denominator of 3 and 27 is 27.
05

Rewrite the Fractions with a Common Denominator

Rewrite \(\frac{1}{3}\) as \(\frac{9}{27}\), so that both fractions have a denominator of 27: \[\frac{9}{27} - \frac{1}{27}\].
06

Perform the Subtraction

Subtract the numerators: \[\frac{9}{27} - \frac{1}{27} = \frac{8}{27}\].
07

Simplify if Possible

Check if \(\frac{8}{27}\) can be simplified. Since 8 and 27 have no common factors (other than 1), \(\frac{8}{27}\) 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.

Reciprocal
Negative exponents can be tricky, but they represent a simple concept. When you see a negative exponent, it means you need to find the reciprocal of the base.
The reciprocal of a number is essentially 1 divided by that number. For example, if we have a number like 3, the reciprocal would be \( \frac{1}{3} \). It's just flipping the number to turn it into a fraction.
  • For instance, with a negative exponent: \( 3^{-1} \) becomes \( \frac{1}{3^1} \), which simplifies to \( \frac{1}{3} \).
  • Similarly, \( 3^{-3} \) translates to \( \frac{1}{3^3} \), making it \( \frac{1}{27} \).
Understanding this is crucial for simplifying expressions with negative exponents, and allows for proper evaluation of expressions like the one given in the exercise.
Fractions
Fractions represent parts of a whole. They consist of a numerator (the top part) and a denominator (the bottom part). In problems involving subtraction of fractions, understanding how they operate helps simplify the process.
For fractions like \( \frac{1}{3} \) and \( \frac{1}{27} \), you need to focus on their numerators and denominators. Here:
  • \( \frac{1}{3} \) has a numerator of 1 and a denominator of 3.
  • \( \frac{1}{27} \) has a numerator of 1 and a denominator of 27.
To subtract these fractions, the numerators will change while the denominators remain the same once a common denominator is found. Fractions should be managed carefully to avoid mistakes in arithmetic operations.
Common Denominator
Subtraction of fractions requires a common denominator. This is a shared denominator that allows you to subtract the numerators directly.
In our exercise, when we have fractions like \( \frac{1}{3} \) and \( \frac{1}{27} \), they do not share a common denominator by default. The least common denominator (LCD) is the smallest multiple both denominators share. For 3 and 27, the LCD is 27 because 27 is a common multiple for both:
  • Convert \( \frac{1}{3} \) to a form with the denominator 27. To do this, multiply both numerator and denominator by 9 to get \( \frac{9}{27} \).
With both fractions now sharing the denominator of 27, you can subtract: \( \frac{9}{27} - \frac{1}{27} = \frac{8}{27} \). Finding common denominators is key to managing and simplifying fractions effectively.

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Most popular questions from this chapter

\(77-82\) me Rationalize the denominator. $$ \frac{1}{2-\sqrt{3}} $$

\(89-96\) m State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ \frac{b}{b-c}=1-\frac{b}{c} $$

Write each number in decimal notation. $$ 8.55 \times 10^{-3} $$

Let \(a, b,\) and \(c\) be real numbers with \(a > 0, b < 0,\) and \( c < 0 .\) Determine the sign of each expression. \(\begin{array}{ll}{\text { (a) } b^{5}} & {\text { (b) } b^{10} \quad \text { (c) } a b^{2} c^{3}} \\ {\text { (d) }(b-a)^{3}} & {\text { (e) }(b-a)^{4}} \quad {\text { (f) } \frac{a^{3} c^{3}}{b^{6} c^{6}}}\end{array}\)

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer correct to the number of significant digits indicated by the given data. $$ \left(7.2 \times 10^{-9}\right)\left(1.806 \times 10^{-12}\right) $$
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