You should be pretty confident with simple numerical indices by now. So lets move on to some negative and fractional indices. Negative indices are all exponents or powers that have a minus sign in front of them and are as result negative. They are quite easy to deal with as there is only one thing that you have to do. Just quickly have a look at the example on the right. We start of with 4 to the power of -2. This is equivalent to 4 to the power of -2 over 1. To get rid of the minus, the only thing you have to do is flip the fraction around (or take its reciprocal) and remove the minus in the exponent. Now the exponent is positive and you can easily solve it. The 1 remains on top of your answers. Here are a couple of examples.Fractional IndicesFractional indices are a bit trickier than negative indices. The example on the right shows how they work very well. Both parts of the fractional exponent have a meaning. The bottom number in the fraction stands for the type of root. In this case, we have to find the cube root of 27 since the base (number to which the exponent belongs) is 27 and the bottom number in the fraction is 3. To make it clearer, we tend to put this part of the expression into brackets. The upper number in the fraction stands for the exponent of the solution of the square root operation inside the brackets. The only step remaining is to take this solution which in this example is 3 and square it. Depending on whether one of the numbers in the fractional exponent is 1, you may not have to put a root or an exponent. To make sure that you have understood, we have put a few examples underneath that may help you (ignore the brackets around the fractional exponents).Negative & Fractional IndicesNow that you know how to do negative and fractional indices, we will try to combine them in one problem. There is no large huge change as you just have to apply both procedures to the problem. There will also be one very challenging example. Try to find out what’s going on. (Ignore the brackets around the fractional exponents.)Did you understand this? If no, don’t worry. This is a very advanced exercise. It is a fraction with a negative and fractional exponent. The first step in his example is to flip the fraction around an remove the minus. Then you square root the fraction before calculating it to the power of 4. If you have no problem with this type of expression, you can consider yourself a very accomplished mathematician in the area of fractions and indices.Trying Some SurdsNow that you have studied simple, fractional and negative indices, you can try to do some surds. Surds involves using square roots and relates to the indices. Please share this page if you like it or found it helpful!
Example 1In this example, we simply flipped the fraction around which allowed us to make the exponent positive so that we can easily find the answer.
Example 2In example 2, we also simply used the reciprocal and then solved the expression with the positive exponent.
Example 1In example 1: we have to square root 16 and then cube our solution to get the answer: 64.
Example 2In example, 2, we have to find the cube root instead of the square root. The general procedure remains the same.
Example 3In the last example we have to find the fourth root. We can see this by looking at the bottom number of the fractional exponent.
Example 1In example 1, we start off by taking the reciprocal and square rooting 9. Then we cube the answer and get 1/27 as the final solution.
Example 2In example 2, we do the same procedure as in example one just that we cube root and square.
Example 3 - CHALLENGE!Did you understand this? If no, don’t worry. This is a very advanced exercise. It is a fraction with a negative and fractional exponent. The first step in his example is to flip the fraction around an remove the minus. Then you square root the fraction before calculating it to the power of 4. If you have no problem with this type of expression, you can consider yourself a very accomplished mathematician in the area of fractions and indices.
ULTIMATE MATHSBecoming an Accomplished MathematicianUltimate Maths is a professional maths website that gives students the opportunity to learn, revise and apply different maths skills. We provide a wide range of lessons and resources...
Quality ContentA wide range of quality learning resources is at your disposal.
Effective TeachingExplanations, examples and questions combined for an effective learning experience.
Easy NavigationA simple user interface ensures that you find the topics you are looking for.
Excellent SupportOur fast and reliable support answer all your questions to your satisfaction.
Chapter 7.2: Learning OutcomesStudents will be able to identify and solve negative indices!Students will be able to identify and solve fractional indices!Students will be able to tackle problems involving both negative & fractional indices!