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Chapter 1: Problem 120

Explain why \((-17)^{-8}\) is positive.

Short Answer

Expert verified
Because it simplifies to \(\frac{1}{(17)^8}\), which is positive.

Step by step solution

01

- Understanding Negation and Exponents

First, recall that raising a negative number to a power affects the sign of the result. A negative number raised to an even power results in a positive number.
02

- Analyze the Given Expression

The given expression is \((-17)^{-8}\). Notice that \(-17\) is raised to the power of \-8\.
03

- Work with the Negative Exponent

A negative exponent means taking the reciprocal of the base raised to the positive of that exponent. So, \((-17)^{-8}\) becomes \(\frac{1}{(-17)^8}\).
04

- Simplify the Expression

Now simplify \((-17)^8\). Since \(-17\) is raised to an even power (8), the result is positive. Therefore, \((-17)^8 = (17)^8\).
05

- Conclusion

Since \((-17)^8\) simplifies to a positive value, \((-17)^{-8}\), which is \(\frac{1}{(17)^8}\) is positive.

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Key Concepts

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

negative exponent
A negative exponent turns your base into a fraction. If you see a negative exponent, like in the expression \((-17)^{-8}\), it means you take the reciprocal (or the inverse) of the base. So, \((-17)^{-8}\) becomes \(\frac{1}{(-17)^8}\).
This change flips the base from being in the numerator to the denominator.
Understanding this idea is crucial because it simplifies the calculation and helps predict the sign of the result. Negative exponents might look complicated at first, but once you take the reciprocal and turn the exponent into a positive one, it becomes much more manageable.
reciprocal
The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 5 is \(\frac{1}{5}\).
In the case of negative exponents, taking the reciprocal is vital. It converts the negative exponent into a positive one, making the expression easier to handle.
Let's consider our example again: \((-17)^{-8}\). The negative exponent tells us to take the reciprocal of \(-17\) raised to the positive power 8. So, it changes to \(\frac{1}{(-17)^8}\). This step turns something potentially confusing into something straightforward and easier to work with.
even power
When a number, whether positive or negative, is raised to an even power, the result is positive. For instance, \(-17\) raised to the 8th power means multiplying -17 by itself 8 times: \((-17)^8\).
Here's why it's positive:
  • Negative times negative equals positive.
  • Since 8 is even, all the pairs of negative numbers multiply to a positive result.
Therefore, \((-17)^8\) ends up being positive. Combine this with the earlier step of converting via the reciprocal, and you see that \(\frac{1}{(-17)^8}\) is positive.
All these concepts tie together to help you see why \((-17)^{-8}\) is, indeed, a positive number.

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Most popular questions from this chapter

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