Square Roots

You already know about squaring, 4² = 16, 7² = 49, and so forth.  The opposite, or inverse, of squaring is taking the square root.  The symbol for taking the square root, or any root for that matter, , is called a “radical” and is used like this:  or 

You can take any integer, square it, and end up with a natural number, but it doesn’t work that way with square roots.  Take  for example.  There’s no integer that squares to 5 but there are two ways to handle that.

The first is simply to leave it in radical form and the second is to find the decimal equivalent.  For this, you’ll need a calculator (there is a formula for determining a root but that’s a subject for another day).  Using a calculator, we get:

 ≈ 2.236068


Adding and subtracting radicals is similar to adding and subtracting polynomial terms, in that you can only combine like terms.  You cannot combine 3x and 2y, so also you cannot combine .  You can’t combine 3 and 5x; likewise, you can’t combine .


Don’t assume that expressions with unlike radicals can’t be simplified.  It’s possible that, after simplifying the radicals, the expression actually can be simplified.  For instance:


Here is an important property of square roots:

Write  as the product of two radicals:

Because square roots are flexible with multiplication, you can factor inside a radical.  You can then split the roots according to the factors.  Likewise, when multiplying radicals, you can multiply the terms within the radicals.

Simplify by writing  as one radical:

…which can be further simplified…


The easiest way to factor inside a radical is to take the primes of each term then take out any pairs.  For example:


That was pretty simple but, it’s important that you understand the technique to aviod mistakes.  In the example above we took the square root of 5² thus wound up with 5 outside the radical.  Likewise, we took the square root of x² three times and outside the radical we must take the power of thexs.  So the result is x³ NOT 3x.

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