In the previous section of this unit, we saw that , 2 and 4 are all the same value, and that value is approximately 6.928. But can we verify that these are the same value, without resorting to using a calculator? Yes we can! Consider the following way of manipulating the expression 2:

2 = 
 · = 
= 

They really are the same thing! Can you do the same with 4?

4 = 
 · = 
= 

Sure enough, both of these expressions are identical to . And it's nice to be able to verify that, but ideally, we'd like to be able to work out a reverse process, whereby we start with , and obtain 4, because 4 is the simplest form.

So let's take the number 48 and rewrite it as a product of primes:

=

Now here I get excited, because I notice that the exponent of 2 is even, which means I can take the square root of it:

= 
 · =
2² · = 
4

Don't forget that taking the square root is the same as dividing the exponent by 2, which is why = 2².

Let's try another example. Simplify .

By finding the prime factorization of 45, we determine that:

= =  · = 3.

Things don't always work out quite this nicely - let's try something just a little more complicated. Simplify .

We start by doing a prime factorization on 54:

=

But here we run into a snag - none of our exponents are even. So is there anything we can do? Yes!

We recognize that the exponent 3 is bigger than two, and it's one more than an even number, which leads us to rewrite 3³ like this:

3³ = 3² · 3¹.

= = =  · = 3.

Let's try a couple more examples.

Simplify

288

.

= = =  ·  · = 2² · 3 · = 12

And one more example: 

Simplify

54000

= = =  · = 2² · 3 · 5 · = 60.

Advanced Tip

In this section, you learned how to simplify something like by finding the prime factorization, and then factoring out any perfect squares. It's useful to point out that we can often avoid having to do an entire prime factorization, if we're good at recognizing perfect squares.

For example, when I look at , the first thing I ask myself is, "Can I think of a perfect square that is a factor of 48?" And right away, I think, "Sure! 16 is a factor of 48!" So instead of doing a full factorization, I do this:

= =  · = 4. And since I know that there are no perfect squares (besides 1) that are a factor of 3, I'm now done.

Here's a similar example: Simplify

I know that 128 is divisible by 4, which is a perfect square:

=  · = 2

However, I also know that 32 is divisible by a perfect square (16!) so now I take it one step further:

2 = 2 = 2 · 4 · = 8

1.

Simplify

2.

Simplify

3.

Simplify

4.

Simplify

5.

Simplify

6.

Simplify

7.

Simplify

8.

Simplify