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 =
b. 271/3 =
d. (16)1/4 =
= -2
= 3
=2
RATIONAL EXPONENTS
Exponent (numerator)
x1/2
Base (x)
Root (denominator)
Exponential
Notation
Radical
Notation
x1/2
x2/3
x3/4
RADICALS
am/n
Index
a-m/n
Radicand
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.
x 11
Simplify using the exponent rules. (No decimals, keep as fractions)
1.
2.
5.
4.
7.
3.
(a3b‐6)(a2b0)
8.
6.
9.
1.
5.
6.
9.
7.
12.
2.
3
15.
24 − 3 3
13.
10.
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)