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Simplifying radicals:  Section 2

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Simplifying powers

Factors of the radicand

Fractional radicand

We can identify with the absolute value of x (Lesson 12).


For, when x 0, then


But if x < 0 -- if, for example, x = −5 -- then


because the square root is never negative. (Lesson 26.)  Rather, when x < 0, then



Therefore in general we must write


conforms to the definition of the absolute value.

Example 4.   Compare  ()²  and  .

()² = x.  (Lesson 26.)  For, in order for that radical to be a real number, the radicand x may not be negative.

= x -- only if x 0.  For any value of x, we must write


Simplifying powers

Example 5.   Since the square of any power produces an even exponent --

(a3)2 = a6

-- then the square root of an even power will be half the exponent.

= a3.

As for an odd power, such as a7, it is composed of an even power times a:

a7 = a6a.


= = a3.

(These results hold only for a 0.)

Problem 5.   Simplify each radical.  (Assume a 0.)

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 a)     =  a²   b)     =  a5   c)     =  an
 d)   = = a      e)    =   =   a4
 f)   = = a7      g)   = = an

Note:  '2n' in algebra, as in part c), indicates an even number, that is, a multiple of 2. The variable n typically signifies an integer. We signify an odd number, then, as '2n + 1,' as in part g).

Problem 6.   Simplify each radical.  Remove the even powers.  (Assume that the variables do not have negative values.)

 a)     =   = 2x
 b)     =   = 2x²y3
 c)     =   = 3x4yz²

Factors of the radicand

Problem 7.   True or false?  That is, which of these is a rule of algebra? (Assume that a and b do not have negative values.)

  a)     True. This is the rule, and the only one. The square root of a product is the product of the square roots of each factor.
  b)   = + False. The radicand is not made up of factors, as in part a).
  c)   = a + b. False! The radicand is not made up of factors.
  d)   = a.   True.
  e)   = a + b. True. The radicand is (a + b.

Problem 8.   Express each radical in simplest form.

a)    =  = 2.

To simplify a radical, the radicand must be composed of factors!

b)    =  = 2a

c)    =  = 3b

Fractional radicand

A radical is in its simplest form when the radicand is not a fraction.

Example 6.  The denominator a square number.   When the denominator is a square number, as , then

= 1

In general,

For, a· a = a².

  Example 7.       =       The definition of division
   =   1

Problem 9.   Simplify each radical.

  a)   1
    b)    =   1
    c)    =   2
  d)    =   5
  Example 8.   The denominator not a square number.
        Simplify  .

Solution.   When the denominator is not a square number, we can make it a square number by multiplying it.  In this example, we will multiply it by itself, that is, by 2.  But then we must multiply the numerator also by 2:

  =       =      =   1

Example 9.   Simplify  .

Solution.   The denominator must be a perfect square.  We can make 50 into a square number simply by multiplying by 2.  We can make x a square by multiplying by x.  And y² is already a square.  Therefore,

Example 10.   Simplify  .  (Assume that the variables do not have negative values.)

Solution.   Again, the denominator must be a perfect square.  It must be composed of even powers.  Therefore, make the denominator into a product of even powers simply simply by multiplying it -- and the numerator -- by bc.  Then extract half of the even powers.

Problem 10.   Simplify each radical.  (Assume that the variables do not have negative values.)

  a)     =   1
  b)     =   1
  c)     =   1
  d)     =     =    5 
  e)     =    2 
    f)     =    a² 
   Problem 11.   Show  

A problem that asks you to show, means to write what's on the left, and then transform it algebraically so that it looks like what's on the right.

  Solution.    =   =   = 

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First Lesson on Radicals

Next Lesson:  Multipying and dividing radicals

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