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Integer powers in practice

Solving exercises about properties and graphs of integer power functions.
Hello. Let us see Exercise 1 of the integers powers in practice step. We have to compare these two numbers, minus pi to the minus fourth over e to the minus 3 and minus e to the minus fourth over pi to minus 3. Let us start. First of all, let us try to write down in a more clear way these two numbers. Minus pi to the minus 4 means exactly 1 over minus pi to the 4, and e to the minus 3 means 1 over e to the 3. On the other side, minus e to the minus 4 means 1 over minus e to the fourth, and pi to the minus 3 means 1 over pi to the 3.
And now, what can we say about this first number? Here, you see you have the fourth power of a negative number. And then what do you get? 1 over pi to the 4, and the denominator is 1 over e to the 3. And on the other side, analogously, we have 1 over e to the fourth, and 1 over pi to the 3. Let us continue on this side. And now, of course, we can write this fraction in a more clever way as e to the 3 over pi to the 4, and this other as pi to the 3 over e to the fourth. We have to compare these numbers.
We have to understand if the number on the left is less or equal or greater than the number on the right. Multiplying on both sides by pi to the fourth and by e to the fourth, we get, equivalently,
the following problem: is e to the seventh less, equal, or greater than pi to the 7? Now, the function, which sends x to x to the seventh is strictly increasing. Therefore, since e is less than pi, we get that e to the seventh is less than pi to the 7. Then this is true. Therefore, this is true. Therefore, this is true. Therefore, this is true. And then we have that this is less than this one.

The following exercises are solved in this step.

We invite you to try to solve them before watching the video.

In any case, you will find below a PDF file with the solutions.

Exercise 1.

Compare the numbers (dfrac{(-pi)^{-4}}{e^{-3}}) and (dfrac{(-e)^{-4}}{pi^{-3}}).

Ex 2. [Solved only in the PDF file]

Compute the domain and try to figure out the range of the functions [f(x)=x^{-2}quadtext{and}quad g(x)=x^{-3}.]

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Precalculus: the Mathematics of Numbers, Functions and Equations

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