Transcript square root

P.3 Radicals and
Rational Exponents
Q: What is a radical? What is a rational
number?
A: A “radical” involves a root symbol,
whereas a “rational number” involves a
fraction.
Definition of the Principal Square
Root
• If a is a nonnegative real number, the
nonnegative number b such that b2 = a, denoted
by b = a, is the principal square root of a.
That is: 4=2 (since 2 squared = 4), not –2
(even though (-2) squared also = 4).
Square Roots of Perfect Squares
2
a a
Ex: Simplify (-3)2
Ex: Simplify x2
Ex: Simplify
-32
Ans: 3, |x|, and “not a real number” or 3i
The Product Rule for Square Roots
• If a and b represent nonnegative real
number, then
ab  a b and
a b  ab
• The square root of a product is the product
of the square roots.
Ex: Compare and draw conclusions:
9+16
vs.
9*16
• Ex: Simplify
a. 500
b. 6x3x
c. 108x6y11
Ans: a)
10 5
b)
3x 2
c)
6 x3 y 5 3 y
The Quotient Rule for Square Roots
• If a and b represent nonnegative real
numbers and b does not equal 0, then
a
a

b
b
and
a

b
a
.
b
• The square root of the quotient is the
quotient of the square roots.
Ex: Simplify
100
9
(ans: 10/3)
Example
We can only add radical expressions if they contain “like terms”:
The same number must be under the radical sign (the radicand),
and it must have the same index. Then just like ordinary “like
terms” we add the COEFFICIENTS and KEEP THE “LIKE”
parts the SAME.
Ex: Perform the indicated operation:
4 3
32 3 
Ans:
(4  1  2) 3  3 3
Ex: Perform the indicated operation:
724 + 26 =
Ans: 16 6
Definition of the Principal nth Root of a
Real Number
n
a b
n
means that b  a
• If n, the index is:
even, and a is nonnegative (a > 0) then b is also
nonnegative (b > 0)
Ex: 4
625  5
odd, a and b can be any real numbers with the same sign (+
or -)
Ex:
3
125  5
Q: What would we write if n is even and a is negative?
(Ans: “not a real number”.)
Finding the nth Roots of Perfect
nth Powers
If n is odd, n an  a
If n is even a  a.
n
n
It is only “necessary” to use the absolute value
symbol if you are finding the even root of a
variable (unknown).
Ex: Simplify each of the following:
³(-2)3
Ans: -2
2
(-2)2
3-8x7y11
2x y
2
2x
24
2
416x8y3
5
-x10
32
x

2
33
y
3
xy
2
The Product and Quotient Rules
for nth Roots
• For all real numbers, where the indicated
roots represent real numbers,
n
a b  ab and
n
n
n
n
a n a

, b0
b
b
Q: Do you remember for what operation(s) you may
NOT separate ( or reverse to put together) the
numbers? (A: sum or difference.)
Definition of Rational Exponents
a1 / n  n a .
Furthermore,
1
1
1/ n
a
 1/ n  n , a  0
a
a
The denominator of the rational exponent becomes the
INDEX of the radical expression.
Ex: Simplify the following:
4½
Ans:
2
(-8)(2/3)
4
5 x3 y 2 3 2 y or 5 x3 y 2 (2 y )(1/ 3)
(250x9y7)1/3
25x
(125x6)2/3
4
Definition of Rational Exponents
a
m/ n
m
n
m
( a)  a .
n
• The exponent m/n consists of two parts: the
denominator n is the root and the numerator
m is the exponent. Furthermore,
a
m/n

1
a
m/n
.
Example:
Ans:
1
8
2x
Simplify 2(-8x12)-(2/3)
Rationalizing the Denominator
ONE TERM in the denominator: simplify, then multiply by
whatever is needed to make a perfect root (ONE TERM).
Ex:
20
20
15 20 15 4 15




15
3
15
15
15
TWO TERMS in the denominator (one is a square root):
simplify, then multiply by the conjugate (TWO TERMS).
Ex:
2
2
2  4 2 2 8



2  16
24
2 4
2 4
2 2 8
2 4
4 2


or
14
7
7
Simplified form for Radical
Expressions:
•NO radical sign in the denominator
•NO fractions under the radical sign
•NO exponents greater than the index under the radical sign
•The index is reduced as low as possible