29
RATIONAL EXPONENTS
THIS SYMBOL 
, as we have seen, symbolizes one number, which is the non-negative 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,
 |
= | 2 | |
| because | |||
| 8 | = | 23. | |
In the symbol ("cube root of 8"), 3 is called the index of the radical. In general,
= b means a = bn.
Equivalently,
If the index is omitted, as in , the index 2 is understood.
| Examples 1. |  |
= | 11. |
 |
= | 2, because 25 = 32. | |
 |
= | 10, because 104 = 10,000. | |
 |
= | −2, because (−2)5 = −32. | |
Thus 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 a
?
To answer, we must preserve the rules of exponents. In particular, we must preserve this rule:
(am)n = amn
Then according to that rule, we will have
(a)² = a· 2 = a1 = a.
But
()² = a.
Therefore we must identify a with .
a =
In general,
a
=
The denominator of a fractional exponent
is equal to the index of the radical.
| Examples 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) | = a |
c) | = (−32) |
Next, what sense can we make of this symbol a
?
According to the rule of multiplying exponents:
a = (a)² = (a²).
That is,
a = ()² = (
).
For example,
8 = (8)² = 2² = 4.
8 is equal to the cube root of 8 squared.
To evaluate a fractional power, it is more efficient to take the root first.
In general,
a
= 
= 
Again:
The denominator of a fractional exponent
indicates the root.
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 raised to a negative exponent has been 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 a |
= | a |
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) | a |
= |  |
b) | a |
= | 1 |
|
| c) | a |
= | 1  |
d) | a |
= | 1  |
|
Evaluations
In the Lesson on exponents, we saw that −24 is a negative number -- it is the negative of 24. (−2)4 is a positive number. 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 |
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 10. 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 11. 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 12. Prove:
=
(ab) =
a· b =
· 
To solve an equation in this form,
| x |
= | b, |
| take the inverse -- the reciprocal -- power of both sides: | ||
(x) |
= | b |
| x | = | b. |
For, x· = x1 = x.
Problem 13. 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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