While geometry is all about visualization, algebra requires you to exercise analytical powers. One of the most interesting of mathematical concepts, that I've personally enjoyed studying, was solving problems based on algebraic exponents. It's an essential piece of algebraic wizardry that you need to master, in order to solve polynomial and numerical problems easily. The focus of this Buzzle article lies in explaining how to solve fractional exponents easily. After a quick roundup of mathematical laws related to solving exponents, I demonstrate the solving of actual examples involving fractional exponents.
What are Exponents?
I assume that having come so far in mathematics, you already know what multiplication is. The idea of an exponent developed out of multiplying the same number with itself, several times. Consider a variable 'm' which is multiplied 5 times with itself. It would be expressed in mathematical form as:
m x m x m x m x m = m
5
Instead of expressing the multiplication of m in such lengthy a form, using a short hand notation, it is expressed in the form - m
5, as 'm' is multiplied 5 times with itself. Here m is called the 'base' and the number 5 is known as 'exponent' or 'power' to which m '
has been raised to'. So m
5 is read as '
m raised to 5' and understood as m multiplied 5 times with itself. Now there are certain rules for multiplying exponents, with the same base term, which are as follows:
| Rules for Solving Exponent Problems |
| ma x mb = m(a + b)
(ma) / (mb) = ma - b
(ma)b = ma x b
m-b = 1/mb
m0 = 1 |
What are Fractional Exponents?
After that brief overview of exponent multiplication laws, let me introduce you to fractional exponents. Just like multiplication of a number with itself can be expressed in terms of exponents, taking square, cube or higher root of a number can also be expressed in exponential form. For example:
√m = m
1/2
3√m = m
1/3
5√(m
2) = m
2/5
Here the terms m
1/2, m
1/3 and m
2/5 have fractional exponents. A fractional exponent is a short hand for expressing the square root or higher roots of a variable. The last of the above terms - 'm
2/5', is 'fifth root of m squared'. Let us take a look at the rules for solving fractional exponents before diving into illustrative examples.
Rules For Solving Fractional Exponents
The rules for simplifying fractional exponents are quite simple. With practice you will find them easier to grasp. Here are the rules which you need to know:
| Rules For Solving Fractional Exponents |
| n√m = m1/n
n√(m)k = mk/n |
These two rules, combined with the ones outlined before, will help you solve exponents based problems quite easily. Let me demonstrate how such problems are solved, through examples, in the following section.
Solving Fractional Exponents?
When it comes to solving mathematical problems, the best way to learn is to study solved examples first and then attempt solving similar examples on your own, in gradually increasing level of complexity. In the following lines, I solve a series of fractional exponent problems to illustrate how it's done.
Example 1: 27
2/3 = (27
1/3)
2 = (
3√27)
2 = 3
2 = 3 x 3 = 9
Example 2: 16
1/4 = (16
1/2)
1/2 = √(√16) = √4 = 2
Example 3: (8
2/3 + 64
4/3) = [(
3√8)
2 + (
3√64)
4] = [2
2 + 4
4] = [4 + 256] = 260
When finding the square root, cube root or higher root of any number, convert it into a multiplicative series of factors. If you are finding the square root, think about which number when multiplied twice, will end up as the given number. That will be its root. For example, if you want to find the square root of 16, think which number when multiplied with itself, gives the answer to be 16. It has to be a number lower than 16. After giving some thought, if you have learned your multiplication tables right, you will realize that 4 is the square root of 16. Similarly, when you want to find the cube root, think which number, when multiplied three times, will provide the given number. With practice, you will find it easier.
If you still find it tough to negotiate them, the only cure is practice. Get hold of a bunch of fractional exponent problems and start solving. Apply the rules presented above and continue solving till you get your answers right. Practice till solving such problems becomes almost second nature to you!
