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Simplifying powers

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.

Therefore,

= = a3.

(These results hold only for a 0.)

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

Do the problem yourself first!

 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

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)2.

Problem 8.   Express each radical in simplest form.

a)    =  = 2.

b)    =  = 2a

c)    =  = 3b

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

 = 12 .

In general,

For, a· a = a2.

 Example 7. = The definition of division = 12

 a)   = 13 b)   = = 13 c)   = = 25
 d)   = = 56

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:

 = = = 12 .

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)   = = 13 b)   = = 15
 c)   = = 17
 d)   = = = 5 6x
 e)   = = 2 x2 f)   = = a2 bc2
 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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Next Lesson:  Multipying and dividing radicals

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