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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

Line 1 is defined by \(y=3 x+7\), and line 2 is perpendicular to line 1. What is the slope of line 2 ? \(\bigcirc-3\) O \(\frac{1}{3}\) \(\bigcirc-\frac{1}{3}\) O I don't know.

If \(x<0\), which of the following is true? C \(x^3\) is positive. ( \(x^3\) is negative. \(x^2\) could be either positive or negative. I don't know.

If \(x<0\), which of the following is true? \(O x^2\) is positive. \(x^2\) is negative. \(x^2\) could be either positive or negative. I don't know.

Even integer q is the product of two distinct prime numbers. What can you say about the factors of q?

In 40 seconds, a mountain biker travels 230 feet. What is her speed in miles per hour? (Note: 1 mile \(=5280\) feet) $$ \begin{array}{r} \frac{230 \times 60 \times 60}{40 \times 5280} \\ \frac{230 \times 5280}{40 \times 60 \times 60} \\ \frac{40 \times 60 \times 60}{230 \times 5280} \\ \text { I don't know. } \end{array} $$
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