Download Math 142 — Rodriguez Lehmann — 4.2

Math 142 — Rodriguez
Lehmann — 4.2
Rational Exponents
I. Define a1/n
A. If the exponent is a whole number:
32 =
20 =
43 =
B. If the exponent is negative:
3–1 =
5–2 =
C. What if the exponent is a fraction? We will show they are roots of a number. The root
of b is another number which when multiplied by itself a given number of times, equals
b.
i.
Square Roots: By definition, a square root of b is a number such that (#)2 = b.
Example: A square root of 9 is ___ since (
)2 =9.
You have seen this written as:
We will show 91/2 is a square root of 9. We will show that 91/2 = ____.
Another square root of 9 is _____.
Note: All numbers have two square roots. We call the nonnegative root the principal
square root.
ii. Cube Roots: By definition, the cube root of b is a number such that (#)3 = b.
The cube root of 8 is ___ because (
)3 = 8.
We will show 81/3 is the cube root of 8. We will show that 81/3 = ____.
Note: There is only one cube root of 8.
Note: The cube root of 8 can be written as 3 8 .
iii. Nth Roots: In general, the nth root of b is a number such that (#)n = b.
The nth root of b is written as ______ and can also be denoted by _______.
Example: 16
¼
=
Examples: Simplify without using a calculator. Then use a graphing calculator to verify
your result.
1) 1001/2
2)
(–64)1/3
3) –811/4
4)
(–81)1/4
321/5
6)
5)
–81/3
Notice:
a. If n is even:
i. and b≥0 then b1/n is a real number. This is the principal nth root of b.
ii. and b < 0, then b1/n is not a real number. Ex: (–4)1/2
b. If n is odd, b can be any real # and b1/n is a real number.
II. Define am/n
A. Let’s look at an example:
43/2 = (41/2)3 =
Likewise,
43/2 = (43)1/2 =
B. In general, am/n =
or
Note: We want a1/n to be a real number so if n is even, a ≥ 0.
Examples: Simplify without using a calculator. Then use a graphing calculator to verify
your result.
1)
Lehmann — 4.2
1252/3
2)
(–64)2/3
Page 2 of 3
−163/2
3)
4)
(–32)3/5
III. Negative Exponents
Recall:
a–m =
1
.
am
Examples: Simplify without using a calculator. Then use a graphing calculator to verify
your result.
1) 16–3/4
2)
(−8)–2/3
3) 81–3/2
4)
–81–3/4
IV. Function Notation
Let f(x)=32x , g(x)=–4(16)x , and h(x)=3(4)x
Find the following:
⎛ 1⎞
5
1) f ⎜ − ⎟
⎝ ⎠
⎛ 3⎞
⎝ 4 ⎟⎠
3) g ⎜
Lehmann — 4.2
⎛ 3⎞
2) h ⎜ ⎟
⎝ 2⎠
4)
⎛ 3⎞
f⎜ ⎟
⎝ 5⎠
Page 3 of 3