Transcript 7.1 nth Roots and Rational Exponents

7.1 nth Roots and Rational
Exponents
2/19/2014
th
n Root
Ex. 32 = 9, then 3 is the square root of 9.
(3 
9)
b
a
If b2 = a, then b is the square root of a.
3
If b = a, then b is the cube root of a.
4
b
3
a
b
4
a
If b = a, then b is the fourth root of a.
n
If b = a, then b is the nth root of a.
You can write the
nth
root of a as
𝑛
𝑎
Where a is a real number and n is the index of the
radical.
Number of Real Roots
𝑛
(use for solving)
n
a
Odd
Any real
number
Even
Greater
than 0
Number
of Roots
Example
3
One
Two
0
One
Less
than 0
No Real
Solution
𝑎
 27   3
3
4
16  2 and  2
2
2
8 2
0 0
 9  no solution
Example 1
𝑛
Find nth Root(s)
𝑎
Find the indicated nth root(s) of a.
a. n = 3, a = – 64
b. n = 4, a = 81
SOLUTION
a. Because n is odd, –64 has one real cube root.
3
– 64 = −4
CHECK ( – 4 )3 = ( – 4 ) ( – 4 ) ( – 4 ) = – 64
b. Because n is even and a is greater than 0, 81 has two
real fourth roots. 4
81 = 3 𝑎𝑛𝑑 − 3
Extra Practice
Find the indicated nth root(s) of a.
1. n = 2, a = 144
ANSWER
12, – 12
2. n = 3, a = 1000
ANSWER
10
3. n = 4, a = 256
ANSWER
4, – 4
Example 2
Solve Equations Using nth Roots
Solve the equation.
a. 2x 4 = 162
SOLUTION
a. 2x 4 = 162
2x 4 162
=
2
2
Write original equation.
Divide each side by 2.
x 4 = 81
4
x4 =
4
x = +
–3
81
Take fourth root of each side.
Example 2
b.
Solve Equations Using nth Roots
−2𝑥 3 = 250
𝑥 3 = −125
3
𝑥3
=
3
x = -5
−125
Divide both sides by -2
Cube root both sides
When to have 1 answer instead of 2
answers when doing problems with
• When the problem says SOLVE, you may have 1 or 2
answers depending if the index is odd or even.
• When the problem says EVALUATE, then you only have
1 answer.
http://www.khanacademy.org/math/algebra/exponent-equations/fractionalexponents-tut/v/basic-fractional-exponents
Vocabulary
Rational Exponents:
exponents written as fractions
Ex :
Radical Form:
a
2
In general:
1
1
1
2
a3
a4
3
4
a 
a
1
a
n

n
a
a
a
Example 3
Evaluate Expressions with Rational Exponents
Evaluate the expression.
a. 91/2 =
9 =3
b. 161/4 =
4
16 = 2
c. 641/3 =
3
64 = 4
d. (– 32 )1/4 =
4
– 32 , no real solution
Extra Practice
Evaluate Expressions
Evaluate the expression.
4. 251/2
ANSWER
5
5. 811/2
ANSWER
9
6. 1251/3
ANSWER
5
321/5
ANSWER
2
7.
http://www.khanacademy.org/math/algebra/exponentequations/fractional-exponents-tut/v/fractional-exponents-withnumerators-other-than-1
Rational Exponents
Ex :
(when numerator is not 1)
3
a2

 a2


1
Radical Form:
In general:
Note:
a
m
5
a4
 a
  a
2
n




3
3
m

 a4


1




5
 a
4
5
n
denominator is the index of the radical and
numerator is the exponent of the radical
Negative Rational Exponent
negative exponent still “moves” power
Ex :

a

3
2
1
a
Radical Form:
In general:

3
a
2
1
 a
a
 a
3

4
n
m


4
1
 a
m
1
5
a4
1
2
5
n
5
Example 4
a. Rewrite
3
( 5)
4
Rewrite Expressions
3
( 5)
4
using rational exponents.
= 53/4
b. Rewrite 72/5 using radicals.
72/5 =
( 5 7 )2
c. Rewrite 2 –2/3 using radicals.
2 –2/3
1
= 2/3
2
=
1
2
( 2)
3
Extra Practice
Rewrite the expression using rational exponents.
8.
9.
( 5 2 )2
1
4
13
ANSWER
22/5
ANSWER
13 –1/4
Extra Practice
Rewrite the expression using radicals.
10.
152/3
11. 11 –1/3
12. 29 – 2/5
ANSWER
ANSWER
ANSWER
(
3
2
15 )
1
3
11
1
(
5
2
29 )
Example 5
Evaluate Expressions with Rational Exponents
Evaluate the expression.
b. 8 –2/3
a. 43/2
SOLUTION
Use radicals to rewrite and evaluate each expression.
a. 43/2 =
b.
8 –2/3
( 4 )3 = 23 = 8
1
= 2/3 =
8
1
( 3 8 )2
1
1
= 2 =
4
2
Checkpoint
Rewrite and Evaluate Expressions with
Rational Exponents
Evaluate the expression.
17. 253/2
ANSWER
125
18. 165/4
ANSWER
32
ANSWER
1
32
19.
8 –5/3
Homework:
WS 7.1
Odd problems only,
skip #11
“I tried to catch some fog. I mist!”