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Chapter 7: Problem 5

$$ \text { If } 0

Short Answer

Expert verified
Given that 0<a<1<b, it can be concluded that 0 < \(a^b\) < 1. This means that \(a^b\) is a fraction and falls in the interval between 0 and 1.

Step by step solution

01

Analyzing the Inequality

The given inequality states that both 'a' and 'b' are positive numbers. Additionally, since \(0< a < 1\), 'a' is a fraction while 'b' is a whole number greater than 1.
02

Understanding Properties of Exponents

Understanding how exponents work is crucial. When a fraction (between 0 and 1) is raised to a positive power, the result is another fraction between 0 and 1. That's because multiplying a fraction results in a smaller number. Conversely, if the base is greater than 1, the exponent will make the number larger.
03

Formulate the Conclusion

Based on the given inequalities and a better understanding of exponent properties, we can conclude that \( 0 < a^b < 1 \). Therefore, \( a^b \) is also a fraction and lies between 0 and 1.

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Key Concepts

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

Inequalities
Inequalities are mathematical statements that compare two values, showing whether one is less than, greater than, or possibly equal to the other. They are particularly useful in identifying ranges and boundaries. In the provided exercise, the inequality \(0 < a < 1 < b\) indicates that 'a' is a fraction between 0 and 1, and 'b' is a number greater than 1. This helps us place both numbers on a number line to better visualize their positions.
  • 'a' is confined between 0 and 1, meaning it could be any fraction such as 0.5, 0.25, or any other number closer to 0 but not equal to it.
  • 'b', being greater than 1, could be 2, 3, or any other number larger than 1.
Understanding these inequalities allows us to anticipate the behavior of numbers within these limits, especially when used as exponents.
Rational Numbers
Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In this exercise, the number 'a' is a rational number, as it falls between 0 and 1, fitting the definition of a fraction.
  • Rational numbers include values like 1/2, 3/4, and 0.2. These are all numbers that can be written as a fraction, even if they appear as decimals.
  • When working with inequalities, rational numbers help us understand where values like 'a' might lie between whole numbers or other rational numbers.
Identifying and working with rational numbers is essential in solving inequality problems, especially when dealing with exponents.
Properties of Exponents
The properties of exponents dictate how numbers behave when used as the power of a base number. Understanding these properties is crucial for solving exercises like the one you're working on.
  • When a number between 0 and 1, like 'a', is raised to a positive exponent greater than 1, it results in a smaller number, still between 0 and 1. This is because \[ a^b = a \times a \times a... \text{(b times)} \] which will shrink since 'a' itself makes smaller products when multiplied by fractions.
  • For a number greater than 1, the exponents amplify or increase the base number. This does not happen in this exercise directly, but helps contrast how bases can behave by themselves or as exponents.
Thus, by applying these properties of exponents, we confirm that if \(0 < a < 1 < b\), then \(0 < a^b < 1\), making \(a^b\) a fractional value as well.

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Most popular questions from this chapter

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