In math, we often learn one set of facts through calculation, and another through related calculations. For example . We can calculate this one. The related fact is that . We know this fact through association rather than calculation. Later, when solving algebraic equations involving a square, we will also need to remember that .
Or how about this, . The associated number fact is that, . We don’t ‘calculate’ that the number multiplied by itself 5 times over to make 32 is 2: we know it from memory.
To help with remembering exponent facts, here is a short list of most commonly used powers: Exponent Cheat Sheet. If you don’t already recall these facts easily, keep the reference sheet handy so that you can focus on the concepts rather than the arithmetic. Here is a long list from wikipedia.
Evaluating positive fractional exponents
Generally, a fractional exponent leads to an irrational number. These questions are called ‘perfect’ because the answers are all integers, and can be evaluated without using a calculator.
We’re asking, what power of four is equal to 16? This is the part that we recall. (For the fourth root you can also do this little trick – square root then square root):
Or just check the Exponent Cheat Sheet.
First, we need to know that . Without knowing that, this question is simply really hard.
This is how we handle it:
We know that .
And we know that
This is why we can rewrite as
In this form, we first evaluate the bracket from memory then calculate the power:
We mean, take the cube root of 125 then square it.
We mean, take the square root of 64 then cube.
Now, isn’t on the cheat sheet. So let’s look for a likely relative. We find:
Now we have it, .
Evaluating negative exponents
Division can be expressed using or by using a fraction or by using a negative exponent. A negative exponent does not make the value negative because dividing does not make a value negative.
1 divided by 7 can be written as:
Note that all of these are positive (all are equal to ). Generally the first thing we do with a negative exponent (which means ‘divide’) is to write it as a fraction (another way to say ‘divide’).
A negative exponent represents repeated division. When we see in algebra, we understand that it means , but the is not necessary to write. Here, it can be helpful to write the 1, as seen in the second line:
Fractions with Negative Exponents
In examples 8, 9 and 10 we take a negative power of a fraction. To understand the process, you need to remember how to divide a fraction. Here’s a reminder:
Watch what happens to the fraction when there is a negative power.
Notice that our orignial fraction flips when the exponent is negative. Let’s write that down in general: