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2.10

## University of Padova

Skip to 0 minutes and 10 secondsHello. 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.

Skip to 1 minute and 35 secondsAnd 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.

Skip to 2 minutes and 37 secondsWe 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,

Skip to 2 minutes and 56 secondsthe 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.

# Integer powers in practice

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 $f(x)=x^{-2}\quad\text{and}\quad g(x)=x^{-3}.$

## Get a taste of this course

Find out what this course is like by previewing some of the course steps before you join:

• ##### Alberto and Carlo explain the course structure
video

Alberto and Carlo explain the course structure

video

Integers

• ##### Rational numbers
video

Rational numbers

video

Real numbers

• ##### Absolute value
video

Absolute value

• ##### An induction proof
video

An induction proof

• ##### The function concept
video

The function concept

• ##### The graph of a function
video

The graph of a function

• ##### Integer powers
video

Integer powers

video

video

• ##### Rational powers
video

Rational powers

• ##### Polynomial and identities
video

Polynomial and identities

• ##### Roots of polynomials
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Roots of polynomials

• ##### Roots of quadratic polynomials
video

Roots of quadratic polynomials

• ##### The Euclidian division algorithm
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The Euclidian division algorithm

• ##### Finding roots
video

Finding roots

• ##### Binomial coefficients
video

Binomial coefficients

• ##### Introduction: types of equations
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Introduction: types of equations

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Equivalence

• ##### Polynomial equations
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Polynomial equations

• ##### Equations involving a radical
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Equations involving a radical

• ##### Equations with several radicals
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Equations with several radicals

• ##### Equations with absolute values
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Equations with absolute values

video

Systems

video

video

video

• ##### Polynomial inequalities
video

Polynomial inequalities

• ##### Inequalities with one radical
video

Inequalities with one radical

• ##### Inequalities with absolute values
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Inequalities with absolute values

• ##### Lines in the plane
video

Lines in the plane

• ##### Systems of linear inequalities
video

Systems of linear inequalities