Showing posts with label square root. Show all posts
Showing posts with label square root. Show all posts

Simplifying Radical Expressions

Now when working with square roots and variables we should be a bit careful.

The variable could represent a positive or negative number so we must ensure that it is positive by making use of the absolute value.
To avoid this technicality many textbooks state, at this point, that we assume all variables are positive. If not, use the absolute values as in the following problems.
Simplify. (Assume variables could be negative.)
To avoid many of the technicalities when working with nth roots we will assume, from this point on, that all the variables are positive.

Simplify. (Assume all variables represent positive numbers.)
Students often try to simplify the previous problem. Be sure to understand what makes the following problems all different.
The property:
says that we can simplify when the operation is multiplication.  There is no corresponding property for addition or subtraction.

Video Examples on YouTube:

Radicals Including Square and Cube Roots

We have seen square roots before, however, we will go into a bit more depth here.  

Also, we will generalize this concept and introduce nth roots. The definition of the principal (non-negative) square root follows.
Not all square roots work out so nicely.  If trying to simplify a square root with a radicand that is not a perfect square, we can find the exact answer using the following properties.
Here A and B are non-negative real numbers and B is not equal to zero. Approximations are made using a calculator.

Give the exact answer and approximate to the nearest hundredth.

This idea can be extended to any positive integer index n.
For 3rd roots, or cube roots, the question is “what raised to the 3rd power will produce the given number?" For example,
Remember that if the radicand of an odd root is negative then the result will be a negative real number.  If the radicand of an even root is negative then the number is not real.
Video Examples on YouTube: