Download MA109, Activity 4: Rational Exponents and Radicals (Section P.4, pp

MA109, Activity 4: Rational Exponents and Radicals (Section P.4, pp.31-35); Date:
Today’s
Goal:
Assignments:
We review the meaning of radicals and rational exponents, that is,
expressions of the form am/n , where the exponent m/n is a rational number.
Homework (Sec. P.4): # 1, 4, 9, 12, 14, 23, 27, 36, 40, 47, 53, 60, 65, 67
(pp. 35-37).
Reading for next lecture: Read Sec. P.5 (pp. 37-41).
√
√
2
The
√ symbol 2 means “the non-negative square root of;” that is a = b means b = a2and b ≥ 0. For instance,
√
16 = 4 as 4 = 16 and 4 ≥ 0 ... but ... there is no real number number b such that b = −16. Thus, −16 is
not defined; that is, we can only take the square root of a non-negative number!
√
Similarly, we can define (when possible) the n-th root of a number a. For instance, 3 8 =
√ b means that we are
looking for the number b that raised to the 3rd power gives 8, i.e., b3 = 8. But 23 = 8, so 3 8 = 2.
◮ Definition of nth root:
If n is a positive integer, then the principal n-th root
of a is the number b that, when raised to the n-th
power, gives a; that is
√
n
a=b
means
bn = a.
Properties of nth Roots:
1.
√
n
2.
r
If n is even we must have a, b ≥ 0.
Example 1: Simplify the expressions below.
When needed, assume the letters denote any real number.
√ √
• 4 24 4 54 =
r
3
•
•
√
3
−
a2 b
a4 b =
We want to define, for instance, a1/3 in a way that is
consistent with the Laws of Exponents. We would like:
a1/3
3
= a(1/3)3 = a1 = a;
thus
a1/3 =
√
3
a
√
√
24 − 8 =
•
p
√
3
64x12 =
•
p
4
√
n
3.
p√
m
n
√
n n ?
a =
√
?
5. n an =
4.
?
a=
if n is odd
if n is even
x4 y 2 z 6 =
Example 2:
Write each radical expression using exponents, and
each exponential expression using radicals.
Radical Expression Exponential Expression
So, by the definition of nth root, we have:
a1/n =
a ?
=
b
•
27
=
8
√
3
n
?
ab =
42/3
a
◮ Definition of Rational Exponents:
For any rational exponent m/n in lowest terms, where
m and n are integers and n > 0, we define
√
am/n = (a1/n )m = ( n a)m
or equivalently
√
am/n = (am )1/n = n am
If n is even we require that a ≥ 0.
7
√
5
53
b−3/2
1
√
x5
Example 3: Simplify the expressions below and eliminate any negative exponents. When needed assume that
all letters denote positive numbers.
Fact:
The Laws of Exponents (review Activity 3)
also hold for rational exponents!
• 10, 000−3/2 =
•
−
27
8
2/3 • 2x4 y −4/5
3
25
64
3/2
8y 2
−1
=
3 x−2 b−1
a3/2 y 1/3
•
3a−2
4b−1/3
•
a2 b−3
x−1 y 2
=
2/3
=
It is useful to eliminate the radical in a denominator.
√
√
3
3
3
5
3 5
3√
=
5.
For instance : √ = √ · 1 = √ · √ =
5
5
5
5
5
5
=
Example 4: Rationalize the denominators:
8
• √
=
5 3
2
This procedure is called rationalizing the denominator.
◮ Rationalizing the Denominator:
√
If the denominator is of the form a, we multiply numer√
ator and denominator√by a. In general, if the denominator is of the form n am with m < n,√then multiplying
n
the numerator and the denominator by an−m will rationalize the denominator.
Example 5 (Drive safely!): Police use the formula
s=
p
30 f d
• p
4
1
3x2 y 3
=
The number f is the coefficient of friction of the
road, which is a measure of the “slipperiness” of the
road. The table gives some typical estimates for f .
to estimate the speed s (in mi/h) at which a car is traveling
if it skids d feet after the brakes are applied suddenly.
Dry
Wet
Tar
1.0
0.5
Concrete
0.8
0.4
Gravel
0.2
0.1
(a) If a car skids 65 ft on wet concrete, how fast was it moving when the brakes were applied?
(b) If a car is traveling at 50 mi/h, how far will it skid on wet tar?
Example 6 (Challenge!):
Without using a calculator, determine which number is larger in each pair:
√
√
3
21/2 or 21/3
5 or 2
Explain!
8