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Chapter 5: Problem 16

Express using positive exponents and, if possible, simplify. $$n^{-6}$$

Short Answer

Expert verified
\[n^{-6} = \frac{1}{n^6}\]

Step by step solution

01

Understand Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal. For example, for any nonzero number 'a' and integer 'n,' we have: \[a^{-n} = \frac{1}{a^n}\]
02

Apply the Rule to the Given Expression

Using the rule from Step 1, rewrite the given expression \(n^{-6}\) as: \[n^{-6} = \frac{1}{n^6}\]

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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 crucial as they represent repeated multiplication. For instance, when we see , it means multiplying the base 'a' by itself 'n' times: . This simplifies many mathematical operations.
Positive exponents are straightforward and simple. They help make complex operations manageable by breaking them down into repeated multiplication.
  • Example: Consider . It means multiplying 2 by itself three times, i.e., = 2 脳 2 脳 2 = 8.
With positive exponents, calculations become simpler, and understanding them makes advanced topics like negative exponents easier.
Simplifying Expressions
Simplifying expressions involves rewriting them in the simplest form without changing their value. This process often uses rules for exponents.
Let's simplify using positive and negative exponents. For example, consider a fraction with exponents like .You should rewrite it as .Here, you move the term with the negative exponent to the denominator to make it positive.
  • First, we identify and apply the rule: .Next, our expression becomes simpler and easier to understand.
  • An important part of simplifying expressions is recognizing patterns and rules. The most common rule concerns converting negative exponents to positive ones as we've shown.
By consistently breaking down expressions, math problems become less daunting.
Reciprocal
The concept of reciprocal is essential when dealing with negative exponents. A reciprocal is simply one divided by the number.
Consider the number 5. Its reciprocal is .
Now, suppose we have . This means we are looking at the reciprocal of 5 squared, which is .
  • In mathematical terms: For a nonzero number 'a', the reciprocal is written as . So, means 1 divided by .
This understanding helps when converting negative exponents to positive exponents. For instance, if we have , it simplifies to by using its reciprocal.
By mastering the concept of reciprocity, handling negative exponents in equations becomes much easier.

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Most popular questions from this chapter

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=6 x^{2}-11 x-10, g(x)=3 x+2 $$

Simplify. $$ \frac{125^{-4}\left(25^{2}\right)^{4}}{125} $$

In computer science, \(1 \mathrm{KB}\) of memory refers to 1 kilobyte, or 1 \(\times 10^{3}\) bytes, of memory. This is really an approximation of 1 \(\times 2^{10}\) bytes (since computer memory uses powers of \(2)\). The TI- 84 Plus graphing calculator has \(480 \mathrm{KB}\) of "FLASH ROM." How many bytes is this?

Find the domain of \(f / g,\) if $$ f(x)=\frac{3 x}{2 x+5} \quad \text { and } \quad g(x)=\frac{x^{4}-1}{3 x+9} $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{3} \square 3^{4} $$
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