Exponent and Radical Rules (6.1, 6.2)
Name: ___________________________________________________
Topic
Definition/Rule
Multiplication
x a ⋅ x b = x a+b
Power to a Power
b
( x a ) = x ab
Power of a
Product
n
(ab ) = a nbn
Zero
x 0 =1
Exponents
x a = x a−b
xb
Division
⎛
⎜
⎝
Power of a
Quotient
1.
Simplifying
2.
3.
n
x ⎞⎟ = x n
y⎠
yn
Day 20
Block: ______________
Example(s)
Simplifying Exponents
Step
Method
1
Label all unlabeled exponents “1”
2
Take the reciprocal of the fraction and make the outside
exponent positive.
3
Get rid of any inside parentheses.
4
Reduce any fractional coefficients.
Move all negatives either up or down.
5
Make the exponents positive.
6
Combine all like bases.
7
Distribute the power to all exponents.
Example
Use properties of exponents to simplify the following expressions.
f.
g.
i.
k.
h.
(51/3
x1/4)3 =
=
121/8
125/6 =
j.
(26
l.
46)-1/6 =
RATIONAL EXPONENTS
QUESTION:
What is the square root of a number?
Determine what number fits into the
.
a.
5 i 5 =
5 is the _____________ of 25
b.
10 i 10 =
10 is the _____________ of 100
c.
x i x =
_____ is the square root of ______
d.
x3 i x3 =
_____ is the square root of ______
e.
x9 i x9 =
_____ is the square root of ______
QUESTION:
x
What does the square root of x mean?
ix
= x1
is the square root of x.
RULE: __________________________
FRACTIONAL EXPONENT RULE:
For any real number a and integers n and m: _____________________________
Examples:
a. 161/2 =
=4
c. (-8)1/3 =
= -2
b. 271/3 =
d. (16)1/4 =
= 3
=2
RATIONAL EXPONENTS
Exponent (numerator)
Exponential
Notation
x1/2
x1/2
x2/3
Base (x)
Root (denominator)
x3/4
RADICALS
Index
am/n
a-m/n
Radicand
Radical Notation
RULE:
EXAMPLES:
Evaluate each of the following without the use of a calculator!
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
RULE:
EXAMPLES:
Evaluate each of the following without the use of a calculator!
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
A negative exponent was slipped into that last problem! How did you deal with it?
RULE:
EXAMPLES:
Evaluate each of the following without the use of a calculator!
1.
5.
9.
2.
6.
10.
3.
4.
7.
8.
11.
12.
Mad Math Minute!
Name: __________________________________
2
______
6.
100
8.
⎛4 3⎞
⎜⎝ x ⎟⎠
= ______
9.
x2=
11.
x5=
______
12.
⎛
⎜⎝
______
14.
16
15.
⎛5 3⎞
⎜⎝ x ⎟⎠
______
17.
⎛ 15 x ⎞
⎝
⎠
18.
4
5.
______
= ______
( )
4
1
y3=
10.
49
13.
x 12 =
16.
5
1
2
x3=
= _____
4.
1
( )
( )
x =
7.
3.
______
1.
3x
= ______
2.
3
x =
Date: __________
2x
3
1
1
1
2
= ____
3
= ______
7
4x
1
2
3
= ______
= ______
______
x 4 ⎞⎟⎠ = ______
3
2
= ______
= ______
RADICALS
Warm Up Simplify the following square root and cube root expressions
1.
3.
2.
4.
EXAMPLE ONE  Simplifying square roots with variables
a)
b)
EXAMPLE TWO  Rationalizing the denominator
a)
5
3
b)
6
7
1+ 2 5
EXAMPLE THREE  Multiplying Radical Expressions
d)
c)
EXAMPLE FOUR  Adding and Subtracting Radical Expressions
a)
b)
c)
PRACTICE
1.
2.
3.
4.
5.
6.
Simplify using the exponent rules. (No decimals, keep as fractions)
1.
2.
5.
4.
7.
8.
3.
(a3b‐6)(a2b0)
6.
9.
x 11
1.
7.
12.
2.
3
24 − 3
3
5.
6.
9.
10.
15.
13.
63 32 − 53 4
5
3+7
4.
14.
2
1− 4 3
EXPONENT PROPERTIES
EXAMPLE FOUR Re-write the radical expression in exponential form.
a)
b)
c)
d)
EXAMPLE FIVE Simplify the expression without using a calculator.
a)
b)
c)
d)