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Simplifying radicals:  Section 2

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Simplifying the square roots of powers

Fractional radicand

 x squared

Simplifying the square roots of 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.

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

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 5.  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 6.     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 7.  The denominator not a square number.  Simplify  radicals.

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

radicals   =   radicals    =   radicals   =   1
2
radicals.

Example 8.   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

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.

end

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First Lesson on Radicals

Next Lesson:  Multipying and dividing radicals

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