Download n th Roots - Algebra 1 and Algebra 2

7.1: nth Roots and Rational Exponents
[Algebra 2 (Y)]
HCPS III:
• Standard 10: Patterns, Functions, and Algebra: PATTERNS AND SYMBOLIC
REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical
situations.
• Benchmark MA.AII.10.8: Add, subtract, multiply, divide, and simplify rational
expressions, radical expressions containing positive rational numbers, and expressions
containing rational exponents.
Goal: Evaluate nth roots of real numbers.
n th Roots
n
Types of Roots
Root
a
Algebra
Square Root
!
“ n th Root of a”
a
n = 2, a = 64;
n = 3, a = 27;
Cube Root
3
a
Fourth Root
!
Fifth Root
!
Sixth Root
!
4
a
5
a
6
a
…
Example
64 = ±8
3
27 = 3
n = 4, a = 16; 4 16 = ±2
!
n = 5, a = 7776; 5 7776 = 6
!
n = 6, a = 4096; 6 4096 = ±4
!
!
!
!
!
Number of Real nth Roots
# of real nth
n
a
roots of a
Odd
Any real #
One
>0
Two
(Pos. #)
Even
0
One!
<0
none !
(Neg. #)
3
Example 1: Find n th Root(s)
Find the indicated nth root(s) of a.
a.) n = 5
4
!
!
!
!
c.) n = 4
d.) n = 2
a = 81
a = "25
!
!
3
"1 = "1
81 = ±3
! 0 = 0;
b.) n = 3
a = 64
a = "32
8 = 2;
4
!
!
!
Example(s)
4
0 =0
"20 , no real #
Example 2: Solve Equations Using nth Roots
Solve the equation.
3
a.) 6x = 384
4
b.) 2x = 162
!
!
5
c.) (x " 8) = 32
3
d.) (x " 3) = 8
!
!
Rational Exponents
Radical Rational Exponent
a
a1 2
3
a
a1 3
4
a!
a!
a1 4
!
a!
a1 6
!
!
!
5
!
6
a1 5
Example 3: Evaluate Expressions with Rational Exponents
Evaluate the expression.
13
a.) 125
12
b.) 16
!
!
c.)
811 4
16
d.) ("64)
!
!
RADICAL
RATIONAL EXPONENT
( )
therefore,
3
2
a
m n
!
"m n
a
!
!
4
13 4
= (2
)
= 24 3
1n m
= (a
=
) = ( a)
1
1n m
(a )
=
n
1
( a)
n
m
m
!
Example 4: Rewrite Expressions
( )
4
3
1
b.) Rewrite 5 7 using
rational exponents.
a.) Rewrite 11 using
rational exponents.
!
c.)
!
35
Rewrite 5 using
radicals.
"2 5
d.) Rewrite 3
radicals.
!
using
!
Example 5: Evaluate Expressions with Rational Exponents
Evaluate the expression.
32
a.) 4
"4 3
b.) 8
!