Negative And Fractional Powers

Negative and Fractional Powers

So, you’ve just got to grips with what powers mean, and everything seems to make sense. For example:

· a³ means a x a x a

· 3² means 3 x 3

And so on. The power indicates the number of times you multiply a number by itself. Easy enough, right? But then you see something absolutely horrible:

64^-2/3

*cue horror music, somebody screaming, a dog howling and so on*

The power is both negative AND a fraction. So, essentially, you’re trying to multiply 64 by itself, negative two thirds times. Which doesn’t make much sense at all.

Before we explain all about this, let’s look at some basic power rules.

· When you multiply a number to a power by that number to a different power, you add the powers

a³ x a⁴ = (a x a x a) x (a x a x a x a), which is seven a’s all multiplied together, which is a⁷. The total number of a’s being multiplied is the sum of the powers.

· When you put a [number to a power] to the power of another number (don’t worry you don’t understand what I mean, it barely made sense to me, and the example will make things clearer), you multiply the powers

(a²)³ = (a x a) x (a x a) x (a x a), which is three lots of two lots of a, which is 3 x 2 a’s, which is 6 a’s all multiplied together, which is a⁶.

Fractional powers

So, let’s say we have a fractional power. To make it easy, we’ll make the numerator (the top number) one. So, we have something like a^1/2.

Let’s do some algebra to help a bit (yes, algebra is hard. Bear with me).

a^1/2 x a^1/2 = a^{1/2 + 1/2} = a¹ = a

So, (a^1/2)² = a

So a^1/2 is the square root of a.

If algebra isn’t your thing (which is okay), just think that two lots of a^1/2 multiplied together gives you an entire a (by adding the powers), so a^1/2 must be (with a bit of deep thinking and some staring into the endless void until it clicks) the square root of a. Because two of them multiplied together make a.

In the same way, a^1/3 is the cube root of a, and a^1/n is the nth root of a (there’s a button on your calculator that can work out nth roots!)

“But wait,” I pretend to hear you cry. “What if the top number isn’t 1?”

Well, this is where our algebra comes in!

a^2/3, for example, is (a^1/3)². So find the cube root of a, then square it! Easy peasy!

For those of you who don’t like algebra, a^2/3 is two lots of a^1/3 multiplied together. So find a^1/3, which is the cube root of a, then times it by itself

We can generalise this concept by saying that a^m/n is the nth root of a multiplied by n (I know that’s a lot of letters and a lot to take in, so if this seems a little iffy to you, go back over the previous stuff until it makes more sense. It’s okay to take things slow, and it’s okay to not understand things the first time around. This is difficult stuff, and you’re doing good if you’ve got this far, so don’t worry if it’s still a bit confusing!)

Negative powers

Yep, you can multiply a number by itself a negative amount of times. Which seems odd, but also sort of makes sense?

When you add a number to something a negative amount of times, you end up doing the opposite operation, which is subtracting that number a positive amount of times. The same applies for powers. If you multiply a number by something a negative amount of times, you end up dividing by the number instead.

Remember that the power a number is raised to indicates how many times you’re multiplying 1 by the number.

· 4² = 1 x 4 x 4

· 4¹ = 1 x 4

· 4⁰ = 1

· 4^-1 = 1/4

· 4^-2 =(1/4)/4

And so on.

If you’d prefer to look at this algebraically (or if you want to try to understand the algebra as well as the wordy explanation):

a⁰ x a¹ = a⁰⁺¹ = a¹, so a⁰ must equal 1.

a^-1 x a¹ = a⁰

Divide both sides by a to get a^-1 = 1/a

Similarly, a^-n = 1/aⁿ

Combining the two

When we combine fractional and negative powers, we deal with the negative first. Using the rule for a [number to a power] to the power of a number, you multiple the powers.

(a^-m/n) = (a^m/n)^-1, which is 1/(a^m/n)

So you create a fraction. Just put one over any number to a power to get rid of the negative sign from the power. Easy peasy. Then just deal with the fractional power as shown above.

Our example

The cube root of 64 is 4, and 4 squared is 16.

64^-2/3 = 1/(64^2/3)

64^2/3 = 4² = 16

So, 64^-2/3 = 1/16

And there you have it! Powers may look tricky, but they’re not too bad once you get used to them. Good luck with your future maths adventures, I believe in you!