Download 10.3 Simplified Form for Radicals

10-1: Rational Exponents
n
If bn = a, then b is an nth root of a and b = a1/n = a , where a is called the radicand and n is
called the root index. If n is a positive even integer and a is a positive real number, then a1/n is
the positive real nth root of a and is called the principal root. If n is a positive odd integer and a
is any real number, then a1/n is the real nth root of a. If m and n are positive integers, then am/n =
(a1/n)m = (am)1/n, provided that a1/n is defined. . If m and n are positive integers, then a -m/n
= m1/n , provided that a1/n is defined and nonzero. It is necessary to use absolute value when
a
taking even roots of variables to indicate the principal root. Product rule: ab  a b .
n
Quotient rule: n a  a . A radical is simplified if it has (1) no perfect nth powers as factors of
n
n
b
n
n
b
the radicand (meaning no powers inside the radicand are larger than the root index), (2) no
fractions inside the radical, and (3) no radicals in the denominator.
Convert from radical to exponential or from exponential to radical form:
1.
3
27
2.
a3
3.
3
w 27
4. -71/2
5. x-2/5
Simplify (assume all variables represent non-negative real numbers)
6.
7.
8. 1691/2
9. -161/4
64
b 36
 1
11.   
 8
1
3
12. (-27)1/3
243 
17. 

 32 
9 1/3
16. (w )
21. (a1/2b-1/3)4/3(a5b)
13. 36m
-3 5
14. 242
18. (16a8b4)1/4
22. (9x8y-10z12)1/2
10
25.
-6
y -15 z 5 
y 9 z -12 
3
5
-2 3
15.
19. 9-191/2
23. 3 81a
1000
 32x
64x
10. (-16)1/4
24.
3
48
20. 32/392/3
 144a8 


18
 y

1
2
10.2 More Expressions Involving Rational Exponents
Multiplying, dividing, and factoring expressions with rational exponents follows the laws of
exponents: [1] To multiply like bases, add the exponents (aman = am+n); [2] To divide like
bases, subtract the exponents  a  a  ; [3] To raise a power to a power, multiply the
m
m n
n
a

exponents (an)m = amn); a negative exponents indicates a reciprocal
 n
1
1
an  .
 a  an and a1  1 


Also,
rational expressions follow the same rules as expressions with integer exponents.
Multiply:
4
4
2
7
7
3
3
1
1
4
12
1. x 5 x 5  x 5
2. 2x 5 y 3 4x 3  3y 3
3. y 2  7 3 y 2  7 3
4.

3z
4
3

 5 2z
7. Divide:
14x
3
7
2
5




3

 35x
7x
2
5.
2

7x
1
2
 3y
5

2
1
2
6.
2
 4x
8. Factor 25x 5 y
5


4
3
1
3

 8 16x
7
2
3
from 75x 5 y
4
 32x
3
1
3
2

 64
 50x 5 y
8
3
Factor each of the following as if it were a polynomial:
3
2
1
4
2
7
9. x 3  x 3  56
10. 16x 9  40x 9  25
11. 15  x  5  4  25  x  5  4
Simplify each of the following to a single fraction:
12.
5
x
1
2
x
3
2
13.
x4
 x  2
1
3
  x  2
2
3

10.3 Simplified Form for Radicals
Property 1:
n
ab  n a n b . Property 2:
n
a

b
n
a
n
a
, b  0 . Property 3: If a is a real number,
then n a n  a if n is even and n a n  a if n is odd. A radical is in simplified form if: [1] none of
the factors of the radicand can be written with an exponent larger than the root index; [2] there
are no fractions under the radical symbols; and [3] there are no radicals in the denominator.
Simplify:
1. 96
2. 3 320
3. 63w15 z 7
4.
7.
5
4
4
10.
x 12 y 5 z 3
2
27
m 3n 5
m 5n
5.
8.
11.
6
5
6.
7
18
9.
3
3
5 xy 2
18 x 2y
7
3
3
12.
x2
a
4
20
2y7