# Examples: Perfect Fractional Exponents

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.

Example1:

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.

Example 2

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:

Example 3:

We mean, take the cube root of 125 then square it.

Example 4

We mean, take the square root of 64 then cube.

Example 5:

Now, isn’t on the cheat sheet. So let’s look for a likely relative. We find:

Now,

And,

Therefore, .

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’).

Example 6:

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:

Example 7:

# 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:

Example 8:

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:

In general,

Example 9

Example 10

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