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28

MULTIPLYING AND DIVIDING
RADICALS

Conjugate pairs

HERE IS THE RULE for multiplying radicals:

multiply radicals

It is the symmetrical version of the rule for simplifying radicals. It is valid for a and b greater than or equal to 0.

Problem 1.   Multiply.

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   a)   multiply radicals· multiply radicals = multiply radicals   b)  2multiply radicals· 3multiply radicals = 6multiply radicals
 
   c)   multiply radicals· multiply radicals = multiply radicals   = 6 d)  (2multiply radicals)2 = 4· 5 = 20
   e)   multiply radicals = multiply radicals    The difference of two squares

Problem 2.   Multiply, then simplify:

multiply radicals

multiply radicals

Example 1.   Multiply  (multiply radicals + multiply radicals)(multiply radicalsmultiply radicals).

Solution.   The student should recognize the form those factors will produce:

The difference of two squares

(multiply radicals + multiply radicals)(multiply radicalsmultiply radicals) = (multiply radicals)2 − (multiply radicals)2
 
  = 6 − 2
 
  = 4.

Problem 3.   Multiply.

a)   (multiply radicals + multiply radicals)(multiply radicalsmultiply radicals)  =  5 − 3 = 2

b)   (2multiply radicals + multiply radicals)(2multiply radicalsmultiply radicals)  =  4· 3 − 6 = 12 − 6 = 6

c)   (1 + multiply radicals)(1 − multiply radicals)  =  1 − (x + 1)  =  1 − x − 1  =  x

d)   (multiply radicals + multiply radicals)(multiply radicalsmultiply radicals)  =  ab

Problem 4.   (x − 1 − multiply radicals)(x − 1 + multiply radicals)

a)   What form does that produce?

The difference of two squares.  x − 1 is "a." multiply radicals is "b.">

b)  Multiply out.

(x − 1 − multiply radicals)(x − 1 + multiply radicals) = (x − 1)2 − 2  
 
  = x2 − 2x + 1 − 2, on squaring the binomial,
 
  = x2 − 2x − 1  

Problem 5.   Multiply out.

(x + 3 + multiply radicals)(x + 3 − multiply radicals) = (x + 3)2 − 3
 
  = x2 + 6x + 9 − 3
 
  = x2 + 6x + 6

Dividing radicals

multiply radicals

For example,

multiply radicals
 multiply radicals
= multiply radicals = multiply radicals

Problem 6.   Simplify the following.

   a)   multiply radicals
multiply radicals
  =  multiply radicals   b)    multiply radicals
8multiply radicals
  =   3
4
multiply radicals   c)    multiply radicals
 multiply radicals
  =   amultiply radicals   =   a· a  =  a2

Conjugate pairs

The conjugate of  a + multiply radicals  is  amultiply radicals.  They are a conjugate pair.

Example 2.   Multiply  6 − multiply radicals  with its conjugate.

Solution.   The product of a conjugate pair --

(6 − multiply radicals)(6 + multiply radicals)

-- is the difference of two squares.  Therefore,

(6 − multiply radicals)(6 + multiply radicals)  =  36 − 2 = 34.

When we multiply a conjugate pair, the  radical vanishes and we obtain a rational number.

Problem 7.   Multiply each number with its conjugate.

a)   x + multiply radicals    multiply radicals = x2 − y

b)   2 − multiply radicals    (2 − multiply radicals)(2 + multiply radicals) = 4 − 3 = 1

  c)   multiply radicals + multiply radicals You should be able to write the product immediately:  6 − 2 = 4.

d)   4 − multiply radicals   16 − 5 = 11

Example 3.   Rationalize the denominator:

    1    
multiply radicals

Solution.   Multiply both the denominator and the numerator by the conjugate of the denominator; that is, multiply them by 3 − multiply radicals.

    1    
multiply radicals
= multiply radicals
 9 − 2
= multiply radicals
    7

The numerator becomes 3 − multiply radicals.  The denominator becomes the difference of the two squares.

  Example 4. multiply radicals = multiply radicals
     3 − 4
= multiply radicals
       −1
 
  = −(3 − 2multiply radicals)
 
  = 2multiply radicals − 3.

Problem 8.   Write out the steps that show the following.

  a)          1     
multiply radicals
  =  ½(multiply radicals)
        1     
multiply radicals
  =   multiply radicals
  5 − 3
  =   multiply radicals
     2
  =   ½(multiply radicalsmultiply radicals)
   The definition of division
  b)         2    
3 + multiply radicals
  =  ½(3 − multiply radicals)
       2    
3 + multiply radicals
  =   multiply radicals
  9 − 5
  =   multiply radicals
      4
  =   ½(3 − multiply radicals)
  c)         _7_    
3multiply radicals + multiply radicals
  =   multiply radicals
     6
       _7_    
3multiply radicals + multiply radicals
  =   multiply radicals
  9· 5 − 3
  =   multiply radicals
      42
  =   multiply radicals
      6
  d)    multiply radicals
multiply radicals − 1
  =   3 + 2multiply radicals
  multiply radicals
multiply radicals − 1
  =   multiply radicals
  2 − 1
  =   2 + 2multiply radicals + 1,   Perfect square trinomial
 
    =   3 + 2multiply radicals
  e)    multiply radicals
1 + multiply radicals
  =   multiply radicals
           x
  multiply radicals
1 + multiply radicals
  =   multiply radicals
1 − (x + 1)
 
    =   multiply radicals
        1 − x − 1
Perfect square trinomial
 
    =   multiply radicals
          −x
 
 
    =   multiply radicals
          x
  on changing all the signs.
  Example 5.    Simplify   multiply radicals
 Solution. multiply radicals = multiply radicals   on adding those fractions,
 
  = multiply radicals   on taking the reciprocal,
 
  = multiply radicals
       6 − 5
  on multiplying by the conjugate,
 
  = 6multiply radicals − 5multiply radicals   on multiplying out.
  Problem 9.    Simplify   multiply radicals
   multiply radicals = multiply radicals   on adding those fractions,
 
  = multiply radicals   on taking the reciprocal,
 
  = multiply radicals
    3 − 2
  on multiplying by the conjugate,
 
  = 3multiply radicals + 2multiply radicals   on multiplying out.

Problem 10.   Here is a problem that comes up in Calculus.  Write out the steps that show:

multiply radicals =  −       ____1____      
xmultiply radicals + (x + h)multiply radicals

In this case, you will have to rationalize the numerator.

multiply radicals   =   1
h
·   multiply radicals
 
    =   1
h
·   _____x − (x + h)_____
multiply radicals
 
    =   1
h
·   ____xxh_____
xmultiply radicals + (x + h)multiply radicals
 
    =   1
h
·  _______−h_______
xmultiply radicals + (x + h)multiply radicals
 
    =   −  _______ 1_______
xmultiply radicals + (x + h)multiply radicals

 

end

Next Lesson:  Rational exponents

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