Download Ch. 10 Radical Expressions

Ch. 10 Radical Expressions
Radical Expressions are expressions with square roots in them.
The square root sign (v ¯) is also called the “radical sign.”
The number or variable expression under the radical sign is called the
“radicand.”
The principal square root of a number is a positive number that when
squared becomes the radicand.
49 = 7
Because 72 = 49.
-7 is also a square root of 49 because (-7)2 = 49. But since -7 is not
positive, it is not the principal square root. To indicate the negative
square root you have to attach a negative sign to the radical sign.
− 7 = − 49
Simplifying Square Roots
Radical Expressions are in simplest form when the radicand contains
no factors that are perfect squares.
The Product Property of Square Roots says you can split a radical
expression into its factors. ab = a ⋅ b
18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2
Example 2: Simplify 252
Are there any factors of 252 that are perfect squares?
36 ⋅ 7 = 36 ⋅ 7 = 6 7
Another definition of a square root is the power ½ .
Taking a number or variable expression to the power ½ is the
same as taking the square root.
16 = (16 )
1
2
x = (x )
1
x = (x )
1
2
6
2
6
=4
2
=x
2
= x3
IMPORTANT!!!!!!
If a ≥ 0 and b ≥ 0,
a 2b 2 = ab
HOWEVER,
a 2 + b2 ≠ a + b
WRONG!!
What if the exponent isn’t even? How do you take the ½ th
power of an odd exponent?
Factor the expression into factors with even powers and factors
with odd powers.
Example 3
Simplify:
b 15
=
b 14 ⋅ b
=
b 14 ⋅ b
= b7 b
Simplify:
3 x 8 x 3 y 13
First factor out any perfect squares of the coefficent, and then factor
out even powers of the variables.
= 3 x 4 ⋅ 2 ⋅ x 2 ⋅ x ⋅ y12 ⋅ y
= 3 x 4 x 2 y12 ⋅ 2 xy
= 3 x(2 xy6 ) 2 xy
= 6 x 2 y 6 2 xy
Now you try this one :
Simplify : 3a 28a 9 b18
Example 5
Simplify:
16( x + 5)
= 4( x + 5)
How about this one?
x2 + 2x + 1
= ( x + 1) 2
= x +1
2
10.2 Addition and Subtracting Radical Expressions
Adding and subtracting radical expressions is like combining like terms.
You cannot add radical expressions that have different terms inside the
radical sign.
9 +
4 ≠
9 +
4 =3+2 =5
13
Therefore, if terms have different numbers inside the radical sign and
these radical expressions cannot be simplified any more, then you cannot
combine them.
You can, however, use the distributive property to factor out any like
terms. Simplify:
12 +
=
8
4 ⋅3 +
4⋅2
= 2 3+2 2
= 2
(
3 +
2
)
Simplify:
3
12 − 5
= 3
4 ⋅3 − 5
= 3⋅2
= 6
9 ⋅3
3 − 5 ⋅3
3 − 15
= (6 − 15
= −9
27
3
)
3
3
3
REMINDER:
Remember, when you factor out a perfect square from a radical
expression, be sure to take the square root of it before bringing
it outside of the radical sign.
16 x 2 y ≠ 16 x 2
16 x 2 y = 4 x
y
y
Example 2B: Simplify
2 x 8 y − 3 2 x 2 y + 2 32 x 2 y
= 2x
4 ⋅ 2 y − 3 2 x 2 y + 2 16 ⋅ 2 x 2 y
= 2x ⋅2
= 4x
2 y − 3x 2 y + 2 ⋅4x
2 y − 3x 2 y + 8x
= (4 x − 3x + 8x) 2 y
= 9x
2y
2y
2y