Close your eyes and slowly whisper "P O W E R". What do you envision? Can you sense the energy that surrounds you? Or maybe you are reminiscing about the power feelings you get when cycling up a hill? Perhaps, you are thinking of something much more concrete - for example the power lines near your house, or maybe you are thinking of power struggles among people. How about the 1990 Snap top-hit I've got the power, maybe that comes to mind? How about powers in mathematics?
Let's discuss power in mathematics. If we are dealing with a positive integer to the power of a positive integer, for example 2^5, it is read as "2 to the power 5", which means 2 multiplied by itself, appearing 5 times. It is often written as,
Is it understandable? Check your understanding. Can you calculate 3^2? Yes, it is 9. How about 4^3? It is 64, which is 4 x 4 x 4. Often, if you have something to the power 2, then it is called squared. If it is to the power 3, then it is cubed. These are just names of specific powers. So "3 squared is 9", and "4 cubed is 64".
There are other things to know about power. For example, what if you are taking something to the power of a fraction, or to the power of 0, or to the power of a negative number, or an imaginary number, or an irrational number? However, we won't go into all those details in this post. Instead, we want you to have a feel about the power of basic mathematical formulas, simple but powerful. We are going to explore some basic algebraic rules that are related to powers.
One way to give some power to your math
is understand the origin of rules.
For general rules, we have to work with variables rather than specific numbers. Like this one below, you know how to read it, don't you? It is "a to the power x", written as:
Here the variable a is called the base and the variable x, is called the exponent. The exponent stands for the number of times that the base a appears. Although a and x represent variables, when you want a specific example, you can give them almost any value.
With this in hand, consider the following:
How would you read each of these expressions in words?
The first expression is read as 'a to the power x multiplied by a to the power y'. Can you figure out how many times does a appear?
All together, a appears x+y times. For example, 2^5 x 2^3 = 2^(5+3) = 2^8. We get the following rule:
The product of two powers with the same base
is the base to the power of the sum of the exponents.
The second expression is read as 'a to the power x, and then to the power y'. How many times does a appear?
As you can see, the number of times that a appears is the product of x and y. For example, (2^5) ^ 3 = 2^ (5x3) = 2^15. We get the following rule:
A power of a power is the base to the power of the product of the exponents.
The third expression is read as 'a to the power x multiplied by b to the power x'. How many times does a appear? It appears x times. How many times does b appear? It also appears x times. What can we do with that?
For example, 2^3 x 3^3 = 6^3. We get the following rule:
The product of two powers with the same exponent is the product of the bases raised to the exponent.
The above rules together with a few others appear in this useful Khan Academy video:
That was simple, wasn't it? Now try and close your eyes and whisper "P O W E R" again. Only this time do it very slowly. As you whisper, visualise the first rule, then the second and finally the third. You need to think a bit as you do it. Do you have the power? Try it a few times and you will.