Fractional powers

To consider fractional power functions, we need to use the third power law. Remember that this says

(x^m)ⁿ = x^nm.

To start with we'll just consider fractions of the form 1/n where n is a whole number.

So we'll start off by considering the expression "x to the power a half": x^1/2.

What happens if we raise this to the power 2, i.e. if we square this expression?
By using the third power law we can see that

(x^1/2)² = x¹ = x,

since "x to the power 1" is just x.

This shows that if we square our expression "x to the power a half" then we get simply x. Therefore "x to the power a half" must be the square root of x.

The square root of a quantity is the number which when you square it gives the original quantity.
Similarly the cube root of a quantity is the number which you have to cube to get the original quantity.

For example the square roots of 16 are 4 and -4, since 4 squared is 16 and (-4) squared is 16.

We have found out above that we can write the square root function as the power of a half, so we can rewrite the last sentence as:

16^1/2 = 4 or -4.

Similarly,

25^1/2=5 or -5,

since 5 squared is 25.

So that's given a meaning to the expression: "x to the power a half". Now consider the expression

x^1/3.

In the work on "x to the power a half", we raised the expression to another power so that we ended up with simply x. What power must we raise "x to the power a third" to, to end up with simply x?

Recall that we will multiply the powers together and we want to end up with the power 1.
So the power we raise our expression to must be 3, since 1/3 x 3 =1.
We therefore raise our expression to the power 3, then we can use the third power law to get:

(x^1/3)³=x.

This tells us that when we cube "x to the power a third", we get x.
Therefore the expression "x to the power a third" is the cube root of x.

For example,

8^1/3=2,

because 2 cubed is 8.

Similarly, we know that 3 cubed is 27 so we can write:

27^1/3=3.

This leads us to the general definition about the expression "x to the power 1/n" where n is a whole number, i.e.

(x^1/n)ⁿ=x.

So x^1/n means the nth root of x, i.e. the number that, when multiplied by itself n times, gives x.

For example the number

81^1/4

is the number which when you multiply 4 of them together gives 81. Since 3x3x3x3=81, we can write that

81^1/4=3.

The integer n on the bottom of the fraction can be positive or negative.
Consider the function "x to the power of minus a half":x^-1/2.

Now by the definition of negative powers, a quantity to the power of -1 means "1 over the quantity" and raising to the power a half means taking the square root, so the function f(x) can be rewritten as:

1/(square root of x).

For example,

25^-1/2 = 1/5.

Finally, you may think we haven't covered all fractional powers yet, only those of the form x^1/n, where n is a whole number,
so what about "x to the power two-thirds" for instance?

The answer is that by using the third power law we can see that

x^2/3 = (x^1/3)².

which is the cube root of x, squared.
In this way, all fractional powers can be reduced to some power of a function of the form

x^1/n.

The paragraph above introduces the idea of a "function of a function". Above we first take the cube root of x and then square the result. We can repeat this process, for instance we could now multiply what we've got so far by 4, then raise the result to the power 5! This idea of carrying out one function and then another on the result of the first is very commonly used, and we will see more of it later in the course.