RENAMING SQUARE ROOTS

Square roots (like $\,\sqrt 5\,$) are one of the most popular radicals.
Like all expressions, square roots have lots of different names!
For example, all the following are just different names for exactly the same number: $$ \sqrt{2700} \qquad \qquad \qquad 2\sqrt{675} \qquad \qquad \qquad 6\sqrt{75} \qquad \qquad \qquad 30\sqrt 3 $$ The last name ($\,30\sqrt{3}\,$) is often the preferred name.
It is said to be in ‘simplest form’, because it has no perfect square factors ‘inside’ the square root.
The process of identifying perfect square factors and (correctly) moving them out of the square root is called ‘simplifying the square root’.

For those in a hurry, here's the punchline—look below to see lots of different ways of doing exactly the same problem.
(Lots of details follow this example, if you've got more time!)

$\sqrt{2700} = 30\sqrt 3$ Quickest and easiest, if you happen to see the biggest perfect square ($\,900\,$)
and like to do things in your head.
$\sqrt{2700} = \sqrt{900\cdot 3} = \sqrt{900}\sqrt{3} = \sqrt{30^2}\sqrt{3} = 30\sqrt 3$ You see the factor of $\,900\,$, but like to write down details of the process.
$\sqrt{2700} = \sqrt{100\cdot 27} = 10\sqrt{27} = 10\sqrt{9\cdot 3} = 10\bigl(3\sqrt 3\bigr) = 30\sqrt 3$ You notice that $\,100\,$ is a factor that's a perfect square.
You don't want to waste time worrying if it's the biggest.
After you take out $\,100\,$, the number left is much more manageable!
$$\begin{align} \sqrt{2700} &= \sqrt{9\cdot 300} = \sqrt{9}\sqrt{300} = 3\sqrt{300}\cr &= 3\sqrt{25\cdot 12} = 3\sqrt{25}\sqrt{12} = 3\cdot 5\sqrt{12} = 15\sqrt{12}\cr &= 15\sqrt{4\cdot 3} = 15\sqrt{4}\sqrt{3} = 15\cdot 2\sqrt{3} = 30\sqrt{3} \end{align} $$ A bit long, but gets you to the same place!
$\sqrt{2700} = 10\sqrt{27} = 10\sqrt{9\cdot 3} = 30\sqrt{3}$ This is what Dr. Burns would likely write down.

CONCEPTS FOR SIMPLIFYING A SQUARE ROOT
(or: WRITING A SQUARE ROOT IN SIMPLEST FORM)

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Writing Radicals in Rational Exponent Form

 
 
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7
8 9 10 11 12 13 14
AVAILABLE MASTERED IN PROGRESS

(MAX is 14; there are 14 different problem types.)