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Principle nth Root
The principal nth root of the real number x is denoted by either 1nx or .n x
Verbally
For 0 :x
The principal nth root is positive for all natural numbers n.
For 0 :x
The principal nth root of 0 is 0.
For 0 :x
If n is odd the principal nth root is negative.
Examples inRadical Notation
Examples in Exponential Notation
16 1
216
3 27 1
327
0 01
20 0
3 0 01
30 0
3 64 1
364 5 32
1532
The principal nth root of the real number x is denoted by either 1nx or .n x
Verbally Examples inRadical Notation
Examples in Exponential Notation
If n is even, there is no real nth root. (The nth roots will be imaginary.)
For 0 :x
1
a real number. is not
121
real number. is not a
Principle nth Root
Use the product rule for exponents to simplify the following expressions:
1.1 1
2 2x x
12x is the principal _________________________
_________________________ of x.
Use the product rule for exponents to simplify the following expressions:
2.1 1 1
3 3 3x x x
13x is the principal _________________________
_________________________ of x.
Use the product rule for exponents to simplify the following expressions:
3.1 1 1 1
4 4 4 4x x x x
14x is the principal _________________________
_________________________ of x.
Rational ExponentsFor a real number x and rational numbers m and n :
Algebraically Examples inRadical Notation
Examples in Exponential Notation
If 1nx
is a real number*
1 m mmnn nx x x
or
1m nm mnnx x x
2233
2
8 8
2
4
2
23 377
67
y y
y
22 13 3
2
8 8
2
4
3 22
3 1 77
6
7
y y
y
Rational ExponentsFor a real number x and rational numbers m and n :
Algebraically Examples inRadical Notation
Examples in Exponential Notation
If 1nx
is a real number*
1mn
mn
xx
34
34
3
116
16
1
21
8
34
34
314
3
116
161
16
1
21
8
*If 0x and n is even, then 1nx is not a real number.
Exponent RulesYou should be familiar with the properties of integer exponents. Note that these properties apply to all real exponents, including rational exponents.Let m, n, x, y, xm, xn, ym, and yn be real numbers
Product rule: m n m nx x x Product to a Power: m m mxy x y
Quotient rule: m
m nn
xx
x
Power rule: nm mnx x
Quotient to a Power: m m
m
x x
y y
Negative power:
n nx y
y x
with 0x and 0.y
Simplify each expression. Assume that x and y are positive real numbers.
31. 1 1
3 25 5
22 5
xy x y
x y