Renaming Square Roots
Need some basic understanding of radicals first? Practice with Radicals
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:
$$ \begin{gather} \cssId{s6}{\sqrt{2700}} \qquad \qquad \qquad \cssId{s7}{2\sqrt{675}} \cr\cr \cssId{s8}{6\sqrt{75}} \qquad \qquad \qquad \cssId{s9}{30\sqrt 3} \end{gather} $$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 several different ways of doing exactly the same problem. (Lots of details follow this example, if you've got more time!)
$\cssId{s16}{\sqrt{2700}} \cssId{s17}{= 30\sqrt 3}$ | Quickest and easiest, if you happen to see the biggest perfect square ($\,900\,$) and like to do things in your head. |
$ \begin{align} \cssId{s19}{\sqrt{2700}}\ &\cssId{s20}{= \sqrt{900\cdot 3}}\cr &\cssId{s21}{= \sqrt{900}\sqrt{3}}\cr &\cssId{s22}{= \sqrt{30^2}\sqrt{3}}\cr &\cssId{s23}{= 30\sqrt 3} \end{align} $ | You see the factor of $\,900\,$, but like to write down details of the process. |
$ \begin{align} \cssId{s25}{\sqrt{2700}}\ &\cssId{s26}{= \sqrt{100\cdot 27}}\cr &\cssId{s27}{= 10\sqrt{27}}\cr &\cssId{s28}{= 10\sqrt{9\cdot 3}}\cr &\cssId{s29}{= 10\bigl(3\sqrt 3\bigr)}\cr &\cssId{s30}{= 30\sqrt 3} \end{align} $ | 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} \cssId{s34}{\sqrt{2700}}\ &\cssId{s35}{= \sqrt{9\cdot 300}}\cr &\cssId{s36}{= \sqrt{9}\sqrt{300}}\cr &\cssId{s37}{= 3\sqrt{300}}\cr &\cssId{s38}{= 3\sqrt{25\cdot 12}}\cr &\cssId{s39}{= 3\sqrt{25}\sqrt{12}}\cr &\cssId{s40}{= 3\cdot 5\sqrt{12}}\cr &\cssId{s41}{= 15\sqrt{12}}\cr &\cssId{s42}{= 15\sqrt{4\cdot 3}}\cr &\cssId{s43}{= 15\sqrt{4}\sqrt{3}}\cr &\cssId{s44}{= 15\cdot 2\sqrt{3}}\cr &\cssId{s45}{= 30\sqrt{3}} \end{align} $ | A bit long, but gets you to the same place! |
$ \begin{align} \cssId{s47}{\sqrt{2700}}\ &\cssId{s48}{= 10\sqrt{27}}\cr &\cssId{s49}{= 10\sqrt{9\cdot 3}}\cr &\cssId{s50}{= 30\sqrt{3}} \end{align} $ | This is what Dr. Burns would likely write down. |
Concepts for Simplifying a Square Root (or: Writing a Square Root in Simplest Form)
Perfect Squares
Perfect squares are found by taking the whole numbers ($0, 1, 2, 3, \ldots$) and squaring them.
Thus, $\,9\,$ is a perfect square. Why? It can be written as a whole number, squared: $9 = 3^2\,$.
Also, $\,(123456789)^2\,$ is a perfect square. Why? It's a whole number, squared.
Perfect Squares You Should Know
Here are perfect squares you should easily recognize for the purpose of simplifying square roots:
$2^2 = \bf \large 4$ $3^2 = \bf \large 9$ $4^2 = \bf \large 16$ $5^2 = \bf \large 25$ |
$6^2 = \bf \large 36$ $7^2 = \bf \large 49$ $8^2 = \bf \large 64$ $9^2 = \bf \large 81$ |
$10^2 = \bf \large 100$ $11^2 = \bf \large 121$ $12^2 = \bf \large 144$ $13^2 = \bf \large 169$ |
$15^2 = \bf \large 225$ $25^2 = \bf \large 625$ |
$20^2 = \bf \large 400$ $30^2 = \bf \large 900$ $40^2 = \bf \large 1600$ (and so on) |
Square Roots Undo Squares
For nonnegative numbers, square roots ‘undo’ squares! Here are examples:
- $\,\sqrt{3^2} = 3\,$ (Think: the square root ‘undoes’ the square!)
- $\,\sqrt{(123456789)^2} = 123456789\,$
Here's the general rule: For all $\,x\ge 0\,$, $\sqrt{x^2} = x\,$.
Definition of Square Root
Be careful: $\sqrt{(-3)^2} \ne -3\,$! By definition: for $\,t\ge 0\,$, $\,\sqrt t\,$ is the nonnegative number which, when squared, gives $\,t\,$. Thus, square roots are never negative.
Renaming the Square Root of a Product
As long as the individual parts aren't negative, you can split the square root of a product into pieces:
For $\,a\ge 0\,$ and $\,b\ge 0\,$:
$\sqrt{ab} = \sqrt{a}\ \sqrt{b}$
Getting Rid of a Perfect Square Factor
When at least one of the factors is a perfect square, look what happens:
$$ \begin{align} \cssId{s107}{\sqrt{75}}\ &\cssId{s108}{= \sqrt{25\cdot 3}}\cr &\cssId{s109}{= \sqrt{25}\sqrt{3}}\cr &\cssId{s110}{= 5\sqrt 3} \end{align} $$We started with $\,\sqrt{75}\,,$ which is not in simplest form. Why not? $\,75\,$ has a factor of $\,25\,$, which is a perfect square.
We ended with $\,5\sqrt 3\,,$ which IS in simplest form. Why? $\,3\,$ has no factors (other than $\,1\,$, of course) that are perfect squares.
Naming Convention for Square Roots
When you have a product involving a square root, always write the square root last. This is particularly important when you're handwriting things, and get a bit sloppy with the length of the upper bar on the square root symbol.
For example, write $\,t\sqrt 5\,,$ not $\,\sqrt 5t\,.$ (This is an exception to the ‘write the constant before the variable’ rule.) Why? You always want to be clear about what's inside the square root, and what's not.
Notice how similar $\,\sqrt 5t\,$ and $\,\sqrt{5t}\,$ look! With the square root at the end, there's no possible confusion.
Resist the Temptation to Multiply Things Out!
When you come across something like $\,\sqrt{304^2}\,,$ resist the temptation to multiply it out to get $\,\sqrt{92416}\,$ (and then sit there for a long time, trying to figure out the square root of $\,92416\,$).
Just go: $\sqrt{304^2} = 304\,.$ Done!