Simplifying Square Roots that Contain Variables

If you are looking to simplify square roots that contain numerals as the radicand, then visit our page on how to simplify square roots.

In this lesson, we are going to take it one step further, and simplify square roots that contain variables. 



Square Roots with Just One Variable


When you take the square root of a term that is "squared", your answer is the " base" of that term. Here's what I mean.

The square root of x squared is x.

So, what happens if the variable is raised to a power other than 2? 

The square root of x raised to the fourth power is x squared.

Did you see how we rewrote the radicand but still maintained its value?  Let's look at another example where a variable is raised to the sixth power.

The square root of x to the sixth power is x to the third power.


Now, let's look at a simplifying a square root that contains the product of a number and a variable.



Simplifying the Square Root of the Product of a Number and a Variable.


The square root of 36 x raised to the 8th power is 6 times x raised to the fourth power.

Let's look at another example where we simplify the square root of multiple factors. Take note of the variable z in this example.  It must remain under the square root since it is z to the first power.


More Simplifying Square Roots with Multiple Variables


The square root of 32 x raised to the fourth power, y raised to the 10th power times z is 4 x squared, y raised to the 5th power times the square root of 2z.

One last example.  If you have a variable that is raised to an odd power, you must rewrite it as the product of two squares - one with an even exponent and the other to the first power.  Check out the variable x in this example.


Simplifying Square Roots


The square root of multiple variables and variables that have odd powers.

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