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Chapter 0: Problem 107

If \(x<0\) and \(m \in \mathbb{Z},\) can \(x^{m}\) be positive? If so, give an example.

Short Answer

Expert verified
Yes, for example, \((-2)^2 = 4\).

Step by step solution

01

Understand the Problem

The problem asks if the expression \(x^m\) can be positive when \(x < 0\) and \(m\) is an integer (\(\mathbb{Z}\)). We need to analyze the behavior of the exponentiation.
02

Analyze Negative Base with Different Exponents

When \(x\) is negative and \(m\) is an integer, the sign of \(x^m\) depends on whether \(m\) is even or odd. If \(m\) is even, \(x^m\) will be positive. If \(m\) is odd, \(x^m\) will be negative.
03

Give an Example

Let's consider an example with \(x = -2\) and \(m = 2\). Here, \(x < 0\) and \(m\) is an integer. Calculating \((-2)^2\): \[(-2)^2 = (-2) \times (-2) = 4\] Therefore, \(x^m\) is positive.
04

Conclusion

Based on the example, \(x^m\) can indeed be positive if \(x < 0\) and \(m\) is an even integer.

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Key Concepts

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

Negative Base
Understanding how exponents work with a negative base is crucial for solving this problem. When we say a base is negative, like \(-x\), it means we are dealing with a number less than zero. If we raise a negative base to a power, the result depends on the exponent we use.

For example, let鈥檚 take \(x = -3\) and see how it behaves when raised to different integer powers. The power can either be an even number or an odd number, which brings us to our next concept.
Even and Odd Exponents
The sign of the result when raising a negative base to a power depends on whether the exponent is even or odd. Here鈥檚 how it works:
  • When the exponent is an even number, like 2, 4, or 6, the result will be positive. This happens because multiplying an even number of negative numbers results in a positive number. For example, \((-3)^2 = 9\) or \((-3)^4 = 81\).

  • When the exponent is an odd number, like 1, 3, or 5, the result will be negative. This is because an odd number of negative multiplications results in a negative number. For instance, \((-3)^1 = -3\) or \((-3)^3 = -27\).

Understanding this can help in visualizing why the result of \(x^m\) with a negative \(x\) can be positive or negative.
Integer Exponents
Integer exponents include all whole numbers, both positive and negative, as well as zero. When dealing with integer exponents, a few key points are important:
  • If the exponent is zero, any non-zero base raised to the power of zero is 1, \((x^0 = 1)\).

  • If the exponent is positive, the result is simply the base multiplied by itself as many times as the exponent indicates.

  • If the exponent is negative, the result is the reciprocal of the base raised to the positive version of the exponent, \((x^{-m} = \frac{1}{x^m})\).

Combining these properties with our understanding of negative bases and even or odd exponents allows us to solve problems involving expressions like \(x^m\) with confidence. For instance, given \(x < 0\) and \m\ as an integer, \(x^m\) can indeed be positive if \m\ is even, such as in the example where \(x = -2\) and \m = 2\, resulting in \((-2)^2 = 4\).

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Most popular questions from this chapter

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (\sqrt{x+1}-5)^{2} $$

Multiply and simplify. $$ \left(-6 a^{5} b\right)\left(\frac{1}{3} a^{2} b^{2}\right) $$

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The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. Determine the total expenditure for private health insurance premiums for the year 2008 .

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ (2 v-\sqrt{17})^{2} $$
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