The Opposite of Exponents

In the introduction we talked about Pythagoras theorem and the task of having to find the square root of a number. That involved finding which number, multiplied by itself, resulted in the number we started with. That task was a special case of the opposite of raising to the power. Raising a number to the power of another number, tells you how many times to multiply the first number by itself. The opposite operation is, when given the first number, and told how many times a second number has been multiplied by itself to get the first number, the task of finding the second number. This is known as finding the root of the first number. Multiplying a number by itself is squaring a number, so finding which number has been multiplied by itself is known as finding the square root. The same thing applies to cubing a number and finding the cubic root.

I explained how you go about doing that for the square root in the introduction. Finding any root (as opposed to the square root) is just the same except you need to do more divisions.

We use the \(\surd\) symbol to represent the operation of finding a root, and we add a small number to the left hand side of it to show what type of root we are trying to find. So, to represent the function of trying to find out which number you multiply by itself three times to get one hundred and twenty five, you would use the symbols \(\sqrt[3]{125}\). This is also known as finding the cube root of one hundred and twenty five. Technically to find which number you multiply by itself twice to get four you could use the symbols \(\sqrt[2]{4}\). However that is also trying to find the square root of four, and square roots are very very common. In fact they are so common that we do not use the small two, we just use the \(\surd\) symbol on its own to represent a square root.

As with addition and multiplication, and you may not be surprised, there is another way to perform this operation as a type of powering. Like multiplication and division, what you do is raise the number to the power of the INVERSE of the root you are trying to find. Sounds a bit complicated. Is actually simple. If you want to find the square root of four, you could also write \(4^{\frac{1}{2}}\). To find the cube root of one hundred and twenty five you would write \(125^{\frac{1}{3}}\).

What about raising a number to a non-unit fraction (something that isn't one half or a third or a blahth)? You work out which number was multiplied by itself the bottom number of the fraction times to get the original number to the power of the top number of the fraction. So if you raise something to the power of three halves (that is a three over a two) what you do is take your number and cube it (the 'three' in the exponent) and then work out the square root (the 'two' in the exponent) of the result. I have no idea why this works, and am not going to try and work it out now.

As with negative numbers and subtraction, and fractions with division, this operation also draws our attention to a new type of number. Another name for fractions is rational numbers. This is because one number divided by another is another way of expressing the ratio between the two numbers. So one divided by three is a third, which means that three is three times as big as one. That is a ratio. So a rational number is a fraction representing the ratio between two whole numbers. For instance two could be said to be a rational number because two is twice as big as one, you could represent this as the ratio \(\tfrac{2}{1}\). Can all numbers be represented this way?

The answer to that question is no. The number which, if multiplied by itself, makes two, cannot be written as a ratio between two whole numbers. We have to just leave it written as \(\sqrt{2}\). We call that number an irrational number because it cannot be represented by any fraction no matter how bit the numbers on top and bottom of the fraction actually are.

How do we know that no such number exists? We can work our way through a series of logic arguments the result of which answers the question completely. As long as the starting assumptions hold up, if your logic is good enough you can be said to have proved something mathematically. The proof of the irrationality of the square root of two is one of the oldest proofs that we know of. Nearly two and a half thousand years old. I will come back to this later on, and properly prove it.

One last word, if you remember when we first looked at exponents we say that if you had a number raised to and exponent raised to another exponent, you just multiplied the exponents together. If you weren't convinced before, you can consider this possibility: \(2^{2^{\tfrac{1}{2}}}\). That's two to the power of two to the power of a half. The answer is NOT \(2^{\sqrt{2}}\), but \(2^{2\cdot\tfrac{1}{2}}\) which is just \(2^1\) or \(2\). If you wanted it to be \(2^{\sqrt{2}}\), you would have to use brackets and write \(2^{(2^{\tfrac{1}{2}})}\)

Next we need to see if these opposite operations have an effect on the order in which you have to do them.