The property
Example 1: Simplifying a Radical



because 25•3 is 75, and 25 is a perfect square 

because


because

So we conclude that
We rewrote 75 as
Example 2: Simplifying a Radical in Two Steps
 
 because 
 because

 So far, so good. But wait! We’re not done! 
 There’s another perfect square to pull out! 
 
 
 Now we’re done. 
The moral of this second example is that after you simplify, you should always look to see if you can simplify again .
A secondary moral is, try to pull out the biggest perfect square you can. We could have jumped straight to the answer if we had begun by rewriting 180 as
This sort of simplification can sometimes allow you to combine radical terms, as in this example:
Example 3: Combining Radicals
 
 We found earlier that

 5 of anything minus 2 of that same thing is 3 of it, right? 
That last step may take a bit of thought. It can only be used when the radical is the same. Hence,
So why does
Of course, the process is exactly the same if variable are involved instead of just numbers!
Example 4: Combining Radicals with Variables
 
 Remember the definition of fractional exponents! 
 As always, we simplify radicals by factoring them inside the root... 
 and then breaking them up... 
 and then taking square roots outside! 
 Now that the radical is the same, we can combine. 
Rationalizing the Denominator
It is always possible to express a fraction with no square roots in the denominator.
Is it always desirable? Some texts are religious about this point: “You should never have a square root in the denominator.” I have absolutely no idea why. To me,
However, there are times when it is useful to remove the radicals from the denominator: for instance, when adding fractions. The trick for doing this is based on the basic rule of fractions: if you multiply the top and bottom of a fraction by the same number, the fraction is unchanged. This rule enables us to say, for instance, that
In a case like
What about a more complicated case, such as
The correct trick for getting rid of
Using this formula, we see that
So the square root does indeed go away. We can use this to simplify the original expression as follows.
Example 5: Rationalizing Using the Conjugate of the Denominator
As always, you may want to check this on your calculator. Both the original and the simplified expression are approximately 1.268.
Of course, the process is the same when variables are involved.
Example 6: Rationalizing with Variables
Once again, we multiplied the top and the bottom by the conjugate of the denominator: that is, we replaced
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