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Simplifying radicals:  Section 2

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 x squared

Simplifying powers

Factors of the radicand

Fractional radicand

radicals

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

radicals.

For, when x 0, then

radicals.

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

radicals

radicals  

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

 radicals.

radicals.

Therefore in general we must write

radicals.

radicals conforms to the definition of the absolute value.

Example 4.   Compare  (radicals)2  and  radicals.

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

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

radicals.

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.

radicals = a3.

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

a7 = a6a.

Therefore,

radicals = radicals = a3radicals.

(These results hold only for a 0.)

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

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 a)   radicals  =  a2   b)   radicals  =  a5   c)   radicals  =  an
 d)   radicals = radicals = aradicals      e)   radicals =  radicals =   a4radicals
 
 f)   radicals = radicals = a7radicals      g)   radicals = radicals = anradical

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)   radicals  =  radicals = 2xradicals
 b)   radical  =  radical = 2x2y3radical
 c)   radical  =  radical = 3x4yz2radical

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)  radicals   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)  radical = radicals + radicals False. The radicand is not made up of factors, as in part a).
  c)  radicals = a + b. False! The radicand is not made up of factors.
  d)  radicals = a.   True.
  e)  radicals = a + b. True. The radicand is (a + b)2.

Problem 8.   Express each radical in simplest form.

a)   radicals =  radicals = 2radicals.

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

b)   radicals =  radicals = 2aradicals

c)   radicals =  radicals = 3bradicals

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 radical, then

radicals = 1
2
.

In general,

radicals

For, a· a = a2.

  Example 7.     radicals  =   radicals     The definition of division
 
   =   1
2
radicals

Problem 9.   Simplify each radical.

  a)  radicals 1
3
    b)  radicals radicals  =   1
3
radicals     c)  radicals radicals  =   2
5
radicals
  d)  radicals radicals  =   5
6
radicals

Example 8.  The denominator not a square number.  Simplify  radicals.

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:

radicals   =   radicals    =   radicals   =   1
2
radicals.

Example 9.   Simplify  radicals.

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,

radicals

Example 10.   Simplify  radicals.  (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.

radicals

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

  a)  radicals radicals   =   1
3
radicals   b)  radicals radicals   =   1
5
radicals  
  c)  radicals radicals   =   1
7
radicals
  d)  radicals radicals   =   radicals   =    5 
6x
radical
  e)  radical radical   =    2 
x2
radical     f)  radical radical   =    a2 
bc2
radical
   Problem 11.   Show   radical

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.   radical  =  radical  =  radical  =  radical
end

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First Lesson on Radicals

Next Lesson:  Multipying and dividing radicals

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