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Dividing and Multiplying Radicals

Date: 03/07/99 at 09:31:50
From: Marta
Subject: Radical Multiplication/Division     

I do not understand how to divide and multiply radicals. (e.g. square 
root of 3/4 times square root of 4/5).  Another problem is dividing 
whole numbers and radicals (e.g. 3/(5-square root of 2) and (3 times 
the square root of 7)/(-1 - square root of 27))

Date: 03/07/99 at 14:35:30
From: Doctor Ezra
Subject: Re: Radical Multiplication/Division

First of all, the radicals you are asking about are just numbers that
involve square roots of other numbers. Let us write sqrt to stand for 
'the square-root of'.

Rule 1. If you have two positive numbers x and y, then 

  sqrt(x*y) = sqrt(x)*sqrt(y), and
  sqrt(x/y) = sqrt(x)/sqrt(y).

That is, the square root of the product is the product of the square 
roots, and the square root of the quotient is the quotient of the 
square roots. That means that to figure out the product of the square 
root of a fraction with the square root of another fraction, just 
multiply the fractions and then take their square root.

Here is an example that is a bit different from yours:

 sqrt(5/7)*sqrt(7/11) = sqrt((5/7)*(7/11))
        = sqrt((5*7)/(7*11)) (you can cancel the 7's)
        = sqrt(5/11),

which is as far as you can go. If you want to rationalize 
the denominator, just multiply top and bottom by sqrt(11):

  sqrt(5/11) = sqrt(5/11)*sqrt(11/11)
      = sqrt((5*11)/(11*11)) (note sqrt(11*11) = 11)
      = sqrt(55) / 11

For dealing with fractions with numbers such as 8 - sqrt(3), you have 
to rationalize the denominator first. If the denominator looks like 
x + sqrt(y), then multiply top and bottom by x - sqrt(y). The reason 
this works is that

 (x + sqrt(y)*(x - sqrt(y)) = x*x - sqrt(y)*sqrt(y)
       = x^2 - y,

which does not have any radicals.

So, for example, to figure out what this fraction is:    

 (5 + sqrt(3))/(-7 + sqrt(11)) 

first multiply top and bottom by (-7 - sqrt(11)). That will give you

   ((5 + sqrt(3))*(-7 - sqrt(11)))/((-7)^2 -11)
   = (-35 - 5*sqrt(11) - 7*sqrt(3) - sqrt(33))/(49-11)
   = (-35 - 5*sqrt(11) - 7*sqrt(3) - sqrt(33))/38.

Now you can apply these rules to your problems and get the answers. 

- Doctor Ezra, The Math Forum   
Associated Topics:
High School Square & Cube Roots
Middle School Square Roots

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