Showing posts with label rationalize. Show all posts
Showing posts with label rationalize. Show all posts

Multiplying and Dividing Radical Expressions

As long as the indices are the same, we can multiply the radicands together using the following property.
Since multiplication is commutative, you can multiply the coefficients and the radicands together and then simplify.


Instructional Video: Multiplying Radicals

Take care to be sure that the indices are the same before multiplying.  We will assume that all variables are positive.


Divide radicals using the following property.
Divide. (Assume all variables are positive.)
Rationalizing the Denominator
A simplified radical expression cannot have a radical in the denominator.  When the denominator has a radical in it, we must multiply the entire expression by some form of 1 to eliminate it.  The basic steps follow.

Rationalize the denominator: 
Multiply numerator and denominator by the 5th root of of factors that will result in 5th powers of each factor in the radicand of the denominator.
Notice that all the factors in the radicand of the denominator have powers that match the index.  This was the desired result, now simplify.
Rationalize the denominator.
This technique does not work when dividing by a binomial containing a radical.  A new technique is introduced to deal with this situation.

Rationalize the denominator: 
Multiply numerator and denominator by the conjugate of the denominator.
And then simplify. The goal is to eliminate all radicals from the denominator.

Instructional Video: Dividing Radicals

Rationalize the denominator.
Video Examples on YouTube:

Complex Numbers

Up to this point, we have been working exclusively with real numbers.  Now we shed this limitation and allow for a much broader range of problems.
With these definitions, we have greatly expanded our space of numbers.  Notice that any real number is also a complex number, for example 5 = 5 + 0i.  Here the real part is 5 and the imaginary part is 0.  Next we consider powers of i.

Instructional Video: Complex Number Operations

Adding and subtracting complex numbers is just a matter of adding like terms. Be sure to use the order of operations and add real and imaginary parts separately.

Add or subtract.
We have to use a bit of caution when multiplying complex numbers.  First, we will run into i^2 often. In this case, we will replace them all with -1.  In addition, the property
is true only when either A or B is non-negative; therefore, simplify using the imaginary unit before multiplying.
Tip: Make use of the imaginary unit if the radicand is negative before trying to simplify.

Multiplying complex numbers often requires the distributive property.
Dividing complex numbers requires techniques similar to rationalizing the denominator.

When a complex number is in the denominator, multiply numerator and denominator by its conjugate.

Video Examples on YouTube: