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2.10

Precalculus

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
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Alberto and Carlo explain the course structure

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Integers

• Rational numbers
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Rational numbers

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Real numbers

• Absolute value
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Absolute value

• An induction proof
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An induction proof

• The function concept
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The function concept

• The graph of a function
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The graph of a function

• Integer powers
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Integer powers

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• Rational powers
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Rational powers

• Polynomial and identities
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Polynomial and identities

• Roots of polynomials
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Roots of polynomials

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• The Euclidian division algorithm
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The Euclidian division algorithm

• Finding roots
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Finding roots

• Binomial coefficients
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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

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video

• Equations with absolute values
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Equations with absolute values

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Systems

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• Polynomial inequalities
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Polynomial inequalities

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• Inequalities with absolute values
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Inequalities with absolute values

• Lines in the plane
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Lines in the plane

• Systems of linear inequalities
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Systems of linear inequalities