Solving Radical Expressions

Date: 07/26/98 at 14:03:05
From: sherri
Subject: Radical expressions

I am having problems solving radical expressions. For example, how do 
you solve the cube root of 32 minus the cube root of 108? What about 
2 times the cube root of 125 minus 5 times the cube root of 64?

Please help!

Date: 07/26/98 at 16:49:02
From: Doctor Ken
Subject: Re: Radical expressions

Hi Sherri,

In these problems, let's think about what "cube root" means. The cube 
root of X is a number you can multiply by itself 3 times to get X. So 
the cube root of 8 is 2, since you can multiply 2*2*2 to get 8.

Now let's learn a couple of things about cube roots. To make writing 
easier here, I'll use {X} to mean the cube root of X, {8} to mean the 
cube root of 8, and so on.

Let's say we have the cube root of 72, {72}. We can change this some 
to make it perhaps easier to deal with. Since 72 = 2*2*2*3*3, we can 
rewrite {72}:

   {72} = {2*2*2*3*3}
        = {2*2*2}*{3*3}
        = 2*{9}

When you're trying to pull numbers out of cube roots, you should look 
for these groups of 3 of the same numbers. I factored 72, and noticed 
that there were three 2's in it. So I could pull those out of the cube 
root.

Now let's try your problems. Your first problem is {32} - {108}.  
Let's factor them:
   
   {32} - {108} = {2*2*2*2*2} - {2*2*3*3*3}     (factor)
                = {2*2*2*2*2} - {3*3*3*2*2}     (change the order)
                = {2*2*2}*{2*2} - {3*3*3}*{2*2} (separate)
                = 2*{2*2} - 3*{2*2}       (take cube root of 8 and 27)
                = (2-3)*{2*2}                   (factor out {2*2})
                = -1*{4}                        (simplify the parts)
                = -{4}                          (simplify)

So the cube root of 32 minus the cube root of 108 is equal to the 
negative of the cube root of 4.

In your next problem, you have 2*{125} - 5*{64}. I'll get you started:

   2*{125} - 5*{64} = 2*{5*5*5} - 5*{2*2*2*2*2*2}
                    = ...

Notice that you could also make {64} into {4*4*4} if you wanted to.

Good luck!

- Doctor Ken, The Math Forum
Check out our web site! http://mathforum.org/dr.math/