 |  |
29
RATIONAL EXPONENTS
Exponential form vs. radical form
THIS SYMBOL 
, as we have seen, symbolizes one number, which is the square root of a. By this symbol 
we mean the cube root of a. It is that number whose third power is a.
For example,
because
8 = 23.
In this symbol 
("cube root of 8"), 3 is called the index of the radical. In general,
means a = bn.
Equivalently,
Read 
"The nth root of a."
For example,
-- The sixth root of 64 -- is 2,
because 64 is the 6th power of 2.
If the index is omitted, as in , the index is understood to be 2.
| Examples 1. |  |
= | 11. |
 |
= | 2, because 25 = 32. | |
 |
= | 10, because 104 = 10,000. | |
 |
= | −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 
.
Problem 1. Evaluate each the following -- if it is real.
| a) | 
| = | 3 | b) | 
| = | −3 | c) | 
| = | 2 |
| d) | 
| = | Not real. | e) | 
| = | −5 |
| f) | 
| = | 1 | g) | 
| = | Not real. | h) | 
| = | −1 |
| Problem 2. Prove: |  |
| Hint: Multiply numerator and denominator by |  |
Fractional exponent
What sense can we make of the symbol 
? It turns out that we must identify with .
= .
Why? Because must obey the rules of exponents. And when it does, it obeys the same formal rule that defines , namely
()2 = a.
For, according to the power of a power rule:
()2 = · 2 = a1 = a.
Therefore we must identify with .
In general,
= 
The denominator of a fractional exponent
is equal to the index of the radical.
| Example 2. | 8 means The cube root of 8, which is 2. |
81 means The fourth root of 81, which is 3. |
|
(−32) means The fifth root of −32, which is −2. |
|
8 is the exponential form of the cube root of 8.
is its radical form.
Problem 3. Evaluate the following.
| a) | 9 = 3 |
b) | 16 = 4 |
c) | 25 = 5 |
||
| d) | 27 = 3 |
e) | 125 = 5 |
f) | (−125) = −5 |
||
| g) | 81 = 3 |
h) | (−243) = −3 |
i) | 128 = 2 |
||
| j) | 16.25 = 16 = 2 |
||||||
Problem 4. Express each radical in exponential form
| a) |  = x |
b) | 
| = |  |
c) |
| = | (−32) |
Next, what sense can we make of this symbol a
?
Again, according to the rule of multiplying exponents:
a = (a)2 = (a2).
That is,
For example,
8 = (8)2 = 22 = 4.
8 is equal to the cube root of 8 squared.
Again:
The denominator of a fractional exponent
indicates the root.
Although 8 = (82), 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,
Problem 5. Evaluate the following.
| a) | 27 = (27 )2 = 32 = 9 |
b) | 4 = (4)3 = 23 = 8 |
|
| c) | 32 = (32)4 = 24 = 16 |
d) | (−32) = (−2)3 = −8 |
|
| e) | 81 = (81 )5 = 35 = 243 |
f) | (−125) = (−5)4 = 625 |
|
| g) | 9 = 35 = 243 |
h) | (−8) = (−2)5 = −32 |
|
Problem 6. Express each radical in exponential form.
| a) | = x |
b) |  = x |
c) |  = x |
||
| d) |  = x |
e) |  = x |
f) |  = x |
||
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 |
= | 1 |
= |  |
Problem 7. Express each of the following with a negative exponent.
| a) | 1  |
= | 1 x |
= | x |
b) | 1  |
= | x |
|
| c) | 1  |
= | 1 x  |
= | x |
d) | 1 |
= | x |
|
Problem 8. Express in radical form.
| a) | |
= |  |
b) | |
= | 1 |
|
| c) |  |
= |  |
d) |  |
= |  |
|
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,
−8 is the negative of 8 :
−8 = −22 = −4.
(−8), on the other hand, is a positive number:
(−8) = (−2)2 = 4.
Problem 9. Evaluate the following.
| a) | 9−2 | = | 1 92 |
= | 1 81 |
b) | 9 |
= | 3 | c) | 9 |
= | 1 3 |
| d) | −9
| = | −3 | e) | −92 | = | −81 | f) | (−9)2 | = | 81 | |||
| g) | −9−2 | = | − | 1 81 |
h) | (−9)−2 | = | 1 81 |
i) | −27
| = | −9 | ||
| j) | (−27)
| = | 9 | k) | 27
| = | 1 9 |
l) | (−27)
| = | 1 9 |
| Problem 10. Evaluate |  |
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 |
 |
= | au − v | |
| (ab)u | = | aubu | Power of a product |
| (au)v | = | auv | Power of a power |
 |
= |  |
Power of a fraction |
Example 3. Rewrite in exponential form, and apply the rules.
| |
= | x· x |
| = | x |
|
| = | x |
|
See Skill in Arithmetic, Adding and Subtracting Fractions.
Problem 11. Apply the rules of exponents.
| a) | 4· 4
= 4
= 4
= 2 |
| b) | 8 8 |
= 8
= 8
= 2 |
| c) | (10)
| = 10
= 10−3
=
| 1 1000 |
Problem 12. Express each radical in exponential form, and apply the rules of exponents.
| a) | x
=
x· x
= x
= x |
| b) | x2
=
x2· x
= x
= x |
| c) |  |
= |
(x + 1)
= (x + 1) |
We can now understand that the rules for radicals -- specifically,
-- are rules of exponents. As such, they apply only to factors.
Problem 13. Prove:
=
(ab) =
a· b =
· 
To solve an equation that looks like this:
x |
= | b, |
| take the inverse -- the reciprocal -- power of both sides: | ||
(x) |
= | b |
| x | = | b. |
For, x· = x1 = x.
Problem 14. Solve for x.
| a) | x |
= | 8 | b) | x |
= | −32 | ||||||
| x | = |
8 |
= | 4 | x | = |
(−32) |
= | −8 | ||||
| c) | (x − 1) |
= | 64 | d) | x7 | = | 5 | |||
| x − 1 | = |
64 |
x | = |
 |
|||||
| x | = | 256 + 1 = 257 | ||||||||
| e) | x |
= | 7 | f) |  |
= | 5 | ||
| x | = | 75 | x | = |
5 =
 |
||||
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