NEKG33 Mathematical methods for economists

Chapter 2 : Day 02: Powers, inequalities and sign diagrams

By Lund University

You also did square root-stuff in high school. Come on in if you need to review. Focus today will be on powers, inequalities and sign diagrams.

Powers with rational exponents

This section focuses on powers when the exponents is not necessarily an integer. The main focus is on the case when the exponent is one half which is the square root. We will also look at cube roots and extensions to an arbitrary nth root. From this, we can define a power with an arbitrary rational exponent. It is also important to know which combinations of a base and an exponent is allowed (for example, you cannot take the square root of a negative number if you are working with the real numbers).

Square root

Cube root

Nth root

Powers with rational and real exponent

General powers: Problems

Exercises on square roots and powers

Problem: Square roots

Problem: Square roots

Problem: Square roots and equations

Problem: n’th roots

Problem: Power rules with rational exponents

Problem: Rationalize denominator

Problem: Power functions

Problem: Proving fraction rules

Problem: Fractions, true or false

Inequalities and sign diagrams

We begin this section by looking at some terminology related to inequalities, namely reflexivity, anti-symmetry, transitivity and totalness. We then look at sign diagrams, a helpful method for figuring out the sign of an expression containing a variable. The next topic is double inequalities which are useful for specifying a range of real numbers. We also cover the absolute value of a number in this section, mostly because the definition of an absolute value uses inequalities. Finally, we look at intervals, and concepts related to intervals, such as open and closed intervals, unbounded intervals, boundary points and the interior and the closure of an interval.

Inequalities, Terminology

Sign diagram

Double Inequalities

Absolute Values


Inequalities: Problems

Exercises on inequalities and sign diagrams

Problem: Inequalities

Problem: Solving inequalities

Problem: Sign of expression

Problem: solving inequalities using sign diagrams

Problem: Solving inequalities

Problem: Solving inequalities

Problem: Solving inequalities

Problem: Absolute value

Problem: Absolute value