Rational Exponents

Veena Duraiswamy

 Overview

So far you’ve dealt with exponents in form of integers; this topic will introduce you to rational exponents (fractions). In this chapter, you will learn to simplify expressions that involve numbers raised to rational exponents. Also, solve exponential equations in order to find the exponent value. Standard questions in this chapter are quite simple; all you have to do is find the values of a number with rational exponents and the hardest question in this chapter is simply to find the value of a variable in form of an exponent using factoring. This chapter will give you an opportunity to familiarize yourself with radicals, fractions and gain more confidence in factoring.

Study Tips,  Methods  and or Advice

Things you should already know:

Radicals

 (Not applicable for negative numbers)

Exponent Laws

The New Stuff:

Rational Exponents law

[n is element of natural, x is greater than 1 and n is even.]

When you have a rational exponent simply take the denominator of the exponent and make it your index number of your radical. The numerator becomes the exponent of the radical.

Example:

You can further simplify this:

 Using the same strategy, you have to work backwards. You can express radicals as fractions. The index number becomes the denominator of the rational exponent and the exponent to which the radical is raised to become the numerator of the rational exponent.

Exponential Equations

Once you have mastered how to convert rational exponents to radicals and the other way around, you are ready for the harder stuff, exponential equations. There are four steps involved in solving these equations.

Step 1: Express each term with the same base

Step 2: Simplify the equation using exponent laws (power rule and FOIL)

Step 3: If bases are equal the exponents are equal

Step 4: Solve the rest disregarding the bases using algebra

Taking the next step

To make things a little bit more complicated you can integrate factoring into solving exponential equations. Don’t worry, it’s not that bad. The simplest solution to solving a little bit more complicated exponential equations is variable replacement. Also, you have to remember how to factor complex trinomials, simple trinomials, perfect squares and difference in squares.

Impossible? That’s the reason we introduce factoring to solve exponential equations.

Step 1: Use the power rule to express the power into a quadratic term

Step 2: Use a variable to replace the power of the term that is the

            same in both linear and quadratic term.

Step 3: Now solve the quadratic equation using factoring. Leave the

            constant term alone.

Step 4: Solve each equation individually and

            substitute the variable back to the power

Step 5: Solve the exponential equation. 

Something that will help you familiarize yourself with exponents

Common Powers To Know

Advice

·         is not possible because you can’t take the root of a negative number.  

·        You should know the exponent laws thoroughly

·        Be confident with factoring, if you’re not PRACTICE!

·        When solving your equation, always remember to write that you are substituting values. Example: “Substitute ”

·        It’s simple, as long as you practice and follow the steps

·        Don’t forget to use FOIL when dealing with expanding binomials

·        Deal with negative exponents first

·        Try to avoid using a calculator for exact values

·        Don’t round  up too soon – rounding errors

·        Know the common powers – it’ll save you a whole lot of time

Sample Questions and complete solutions