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29

RATIONAL EXPONENTS

Fractional exponent

 rational exponents

THIS SYMBOL rational exponents, as we have seen, symbolizes one number, which is the square root of a.  By this symbol rational exponents we mean the cube root of a. It is that number whose third power is a.

For example,

 rational exponents

because

8 = 23.

In this symbol rational exponents ("cube root of 8"), 3 is called the index of the radical.  In general,

 

rational exponents   means   a = bn.

 Equivalently,

rational exponents

Read  rational exponents "The nth root of a."

For example,

rational exponents -- The sixth root of 64 -- is 2,
because 64 is the 6th power of 2.

If the index is omitted, as in rational exponents, the index is understood to be 2.

  Examples 1. rational exponents = 11.
 
  rational exponents = 2,   because 25 = 32.
 
  rational exponents = 10,   because 104 = 10,000.
 
  rational exponents = −2,   because (−2)5 = −32.

We see that, if the index is odd, then the radicand may be negative.  But if the index is even, the radicand may not be negative.  There is no such real number, for example, as rational exponents.

Problem 1.   Evaluate each the following -- if it is real.

   a)   rational exponents =   b)   rational exponents = −3    c)   rational exponents =
   d)   rational exponents = Not real.   e)   rational exponents = −5
   f)   rational exponents =  1    g)   rational exponents = Not real.   h)   rational exponents = −1
   Problem 2.   Prove:    rational exponents
  Hint:  Multiply numerator and denominator by  rational exponents

rational exponents

Fractional exponent

What sense can we make of the symbol  rational exponents ?  It turns out that we must identify  rational exponents  with rational exponents.

rational exponents = rational exponents.

Why?  Because rational exponents must obey the rules of exponents. And when it does, it obeys the same formal rule that defines rational exponents, namely

(rational exponents)2 = a.

For, according to the power of a power rule:

(rational exponents)2 = rational exponents· 2 = a1 = a.

Therefore we must identify  rational exponents  with  rational exponents .

In general,

 

rational exponents  =  rational exponents

The denominator of a fractional exponent
is equal to the index of the radical.

  Example 2. 8rational exponents  means  The cube root of 8, which is 2.
 
  81rational exponents  means  The fourth root of 81, which is 3.
 
  (−32)rational exponents  means  The fifth root of −32, which is −2.

8rational exponents is the exponential form of the cube root of 8.

rational exponents is its radical form.

Problem 3.   Evaluate the following.

   a)   9rational exponents  =  3   b)   16rational exponents  =  4   c)   25rational exponents  =  5
 
   d)   27rational exponents  =  3   e)   125rational exponents  =  5   f)   (−125)rational exponents  =  −5
 
   g)   81rational exponents  =  3   h)   (−243)rational exponents  =  −3   i)   128rational exponents  =  2
 
   j)   16.25  =  16rational exponents = 2

Problem 4.   Express each radical in exponential form

   a)   rational exponents  =  xrational exponents   b)   rational exponents = rational exponents   c)   rational exponents = (−32)rational exponents

rational exponents

Next, what sense can we make of this symbol  arational exponents ?

Again, according to the rule of multiplying exponents:

arational exponents  =  (arational exponents)2 = (a2)rational exponents.

That is,

rational exponents

For example,

8rational exponents  =  (8rational exponents)2  =  22 =  4.

8rational exponents is equal to the cube root of 8  squared.

Again:

The denominator of a fractional exponent
indicates the root.

Although  8rational exponents  =  (82)rational exponents, to evaluate a fractional power, it is more efficient to take the root first, because we will be taking the root of a smaller number.

In general,

rational exponents

Problem 5.   Evaluate the following.

   a)   27rational exponents  =  (27rational exponents)2 = 32 = 9   b)   4rational exponents  =  (4rational exponents)3 = 23 = 8
 
   c)   32rational exponents  =  (32rational exponents)4 = 24 = 16   d)   (−32)rational exponents  =  (−2)3 = −8
 
   e)   81rational exponents  =  (81rational exponents)5 = 35 = 243   f)   (−125)rational exponents  =  (−5)4 = 625
 
   g)   9rational exponents  =  35 = 243   h)   (−8)rational exponents  =  (−2)5 = −32

Problem 6.   Express each radical in exponential form.

   a)   rational exponents  =  xrational exponents   b)   rational exponents  =  xrational exponents   c)   rational exponents  =  xrational exponents
 
   d)   rational exponents  =  xrational exponents   e)   rational exponents  =  xrational exponents   f)   rational exponents  =  xrational exponents

Negative exponent

A number with a negative exponent is defined to be the reciprocal of that number with a positive exponent.

a−v  =    1 
av

a−v is the reciprocal of av.

Therefore,

 1 
rational exponents
=  1 
rational exponents
= rational exponents

Problem 7.   Express each of the following with a negative exponent.

   a)     1  
rational exponents
  =    1 
xrational exponents
  =   xrational exponents      b)     1  
rational exponents
  =   xrational exponents
 
   c)     1  
rational exponents
  =    1 
xrational exponents
  =   xrational exponents      d)     1  
rational exponents
  =   xrational exponents

Problem 8.   Express in radical form.

 a)   rational exponents = rational exponents   b)   rational exponents =   1  
rational exponents
 
 c)   rational exponents = rational exponents   d)   rational exponents = rational exponents

Evaluations

In the Lesson on exponents, we saw that −24 is a negative number. It is the negative of 24.

For, a minus sign signifies the negative of the number that follows. And the number that follows −24,  is 24.

[(−2)4 is a positive number.  Lesson 13.]

Similarly, then,

−8rational exponents is the negative of 8rational exponents :

−8rational exponents  = −22  = −4.

(−8)rational exponents, on the other hand, is a positive number:

(−8)rational exponents  =  (−2)2  =  4.

Problem 9.   Evaluate the following.

   a)   9−2   =    1 
92
  =    1 
81
  b)   9rational exponents   =   3   c)   9rational exponents   =   1
3
   d)   −9rational exponents   =   −3   e)   −92   =   −81   f)   (−9)2   =   81
 
   g)   −9−2   =    1 
81
  h)   (−9)−2   =    1 
81
  i)   −27rational exponents   =   −9
  j)   (−27)rational exponents   =   9   k)   27rational exponents   =   1
9
  l)   (−27)rational exponents   =   1
9
  Problem 10.   Evaluate   rational exponents

It is the reciprocal of 16/25 with a positive exponent.
So it is the square root of 25/16, which is 5/4, raised to the 3rd power:  125/64.

The rules of exponents

An exponent may now be any rational number.  Rational exponents u, v will obey the usual rules.

auav = au + v  Same Base
 
rational exponents = au − v  
 
(ab)u = aubu  Power of a product
 
(au)v = auv  Power of a power
 
rational exponents = rational exponents  Power of a fraction

Example 3.   Rewrite in exponential form, and apply the rules.

 rational exponents rational exponents   =   xrational exponents· xrational exponents
 
    =   xrational exponents
 
    =   xrational exponents

See Skill in Arithmetic, Adding and Subtracting Fractions.

Problem 11.   Apply the rules of exponents.

 a)   4rational exponents· 4rational exponents  =  4rational exponents  =  4rational exponents  =  2
 b)   8rational exponents
8rational exponents
 =  8rational exponents  =  8rational exponents  =  2
 c)   (10rational exponents)rational exponents  =  10rational exponents  =  10−3  =     1   
1000

Problem 12.   Express each radical in exponential form, and apply the rules of exponents.

 a)   xrational exponents  =  x· xrational exponents  =  xrational exponents  =  xrational exponents
 b)    x2 
rational exponents  =  x2· xrational exponents  =  xrational exponents  =  xrational exponents
 c)   rational exponents = (x + 1)rational exponents  =  (x + 1)rational exponents

We can now understand that the rules for radicals -- specifically,

rational exponents

-- are rules of exponents.  As such, they apply only to factors.

Problem 13.   Prove:  rational exponents

rational exponents =  (ab)rational exponents = arational exponents· brational exponents rational exponents· rational exponents

rational exponents

To solve an equation that looks like this:

xrational exponents = b,
 
take the inverse -- the reciprocal -- power of both sides:
 
(xrational exponents)rational exponents = brational exponents
 
x = brational exponents.

For,  xrational exponents· rational exponents  =  x1 = x.

Problem 14.   Solve for x.

  a)   xrational exponents = 8     b)   xrational exponents  =  −32  
 
  x  =  8rational exponents = 4   x  =  (−32)rational exponents = −8
  c)   (x − 1)rational exponents = 64 d)   x7  =  5
 
x − 1  =  64rational exponents x  =  rational exponents
 
x  =  256 + 1 = 257  
  e)   xrational exponents  =  7 f)   rational exponents  =  5
 
x  =  75 x  =  5rational exponents  =  rational exponents
end

Next Lesson:  Complex numbers

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