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Simplifying radicals:  Section 2

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Simplifying powers

Factors of the radicand

Fractional radicand

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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

 .

.

This conforms to the definition of the absolute value.

Simplifying powers

Example 4.   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

Therefore,

= = 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

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 formal rule of algebra?  (Assume that the variables do not have negative values.)

  a)     True. This is the rule. 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.  Make factors!

b)    =  = 2a

c)    =  = 3b

Fractional radicand

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

Example 5.  The denominator a square number.   When the denominator is a square number, as , then

= 1
2

In general,

For, a· a = a².

  Example 6.       =       The definition of division
 
   =   1
2

Problem 9.   Simplify each radical.

  a)   1
3
    b)    =   1
3
    c)    =   2
5
  d)    =   5
6
  Example 7.   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
2

Example 8.   Simplify  .  (Assume that the variables do not have negative values.)

Solution.   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
3
  b)     =   1
5
  c)     =   1
7
  d)     =     =    5 
6x
  e)     =    2 
x²
    f)     =    a² 
bc²

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Next Lesson:  Multipying and dividing radicals


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