7.1 nth Roots and Rational Exponents
21
7.1 nth Roots and Rational Exponents 2/19/2014
description
7.1 nth Roots and Rational Exponents. 2/19/2014. n th Root. Ex. 3 2 = 9, then 3 is the square root of 9. If b 2 = a, then b is the square root of a. If b 3 = a, then b is the cube root of a. If b 4 = a, then b is the fourth root of a. - PowerPoint PPT Presentation
Transcript of 7.1 nth Roots and Rational Exponents
7.1 nth Roots and Rational Exponents
2/19/2014
nth RootEx. 32 = 9, then 3 is the square root of 9. If b2 = a, then b is the square root of a.
If b3 = a, then b is the cube root of a.
If b4 = a, then b is the fourth root of a.
If bn = a, then b is the nth root of a.You can write the nth root of a as Where a is a real number and n is the index of the radical.
)93(
ab
3 ab
4 ab
Number of Real Roots(use for solving)
Examplen
Odd
Even
a
Any real number
Greater than 0
0
Less than 0
Number of Roots
One
Two
One
No Real Solution
283 3273
2 and 2164
002
solution no 92
𝑛√𝑎
Example 1 Find nth Root(s)
Find the indicated nth root(s) of a.
a. =n 3, =a 64– b.
=n 4, =a 81SOLUTION
a. Because n is odd, 64 has one real cube root.–
b. Because n is even and a is greater than 0, 81 has two real fourth roots.
CHECK ( )34– = ( )4– ( )4– ( )4– = 64–
3 64–
4 81
𝑛√𝑎
¿− 4
¿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 Write original equation.
=2 2
1622x 4Divide each side by 2.
=x 4 81
=x 3–+
4 =x 4 4 81 Take fourth root of each side.
Example 2 Solve Equations Using nth Roots
b.
Divide both sides by -2
= Cube root both sides
x = -5
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/fractional-exponents-tut/v/basic-fractional-exponents
Ex :
Radical Form:
VocabularyRational Exponents: exponents written as fractions
21
a 31
a 41
a
aa 2 3 a 4 a
In general:nn aa
1
Example 3
Evaluate the expression.
Evaluate Expressions with Rational Exponents
a. 91/2 = 9 = 3
b. 161/4 = 2= 164
c. 641/3 = 4= 643
( )1/432–d. = 324 – , no real solution
Extra Practice
Evaluate the expression.
ANSWER 5
Evaluate Expressions
4. 251/2
5. 811/2 ANSWER 9
6. 1251/3 ANSWER 5
7. 321/5 ANSWER 2
http://www.khanacademy.org/math/algebra/exponent-equations/fractional-exponents-tut/v/fractional-exponents-with-numerators-other-than-1
Ex :
Radical Form:
Rational Exponents(when numerator is not 1)
23
a
32 a 54 a
45
a
In general: nmmn
aa
Note: denominator is the index of the radical and numerator is the exponent of the radical
3
21
a
5
41
a
Ex :
Radical Form:
Negative Rational Exponentnegative exponent still “moves” power
23
a
32
1
a 54
1
a
45
a
In general:
nm
mn
aa 1
23
1
a
45
1
a
Example 4 Rewrite Expressions
( )4 35 = 53/4
Rewrite using rational exponents.( )4 35a.
b. Rewrite using radicals.72/5
72/5 = ( )5 27
=1
( )3 22
Rewrite using radicals. 2 2/3–
2 2/3– =1
22/3
c.
Extra Practice
Rewrite the expression using rational exponents.
8. ( )5 22 ANSWER 22/5
9.1
134ANSWER 13 1/4–
Extra Practice
Rewrite the expression using radicals.
10. 152/3
11. 11 1/3– ANSWER1
113
ANSWER 15( )3 2
12. 29 2/5– ANSWER1
29( )5 2
Example 5 Evaluate Expressions with Rational Exponents
Evaluate the expression.
a. 43/2 b. 8 2/3–
SOLUTIONUse radicals to rewrite and evaluate each expression.
a. 43/2 3= ( )4 = 23 8=
b. 8 2/3– =1
82/3=
1
( )3 28=
122
=41
Checkpoint
Evaluate the expression.
17. 253/2
18. 165/4
19. 8 5/3–
ANSWER 125
ANSWER 32
ANSWER132
Rewrite and Evaluate Expressions with Rational Exponents
Homework: WS 7.1
Odd problems only, skip #11
“I tried to catch some fog. I mist!”
