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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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Most popular questions from this chapter

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