simplifying exponential expressions. exponential notation base exponent base raised to an exponent...
TRANSCRIPT
Simplifying Exponential Expressions
Exponential Notation
BaseExponent
Base raised to an exponent
Example: What is the base and exponent of the following expression?
272 is the
exponent7 is the base
Goal
To write simplified statements that contain distinct bases, one whole number in the numerator and one in the denominator, and no negative exponents.
Ex:
21 4 3 8 4
2 122 1 2
9 9
46
a b b c
aa b c
Multiplying TermsWhen we are multiplying terms, it is easiest
to break the problem down into steps. First multiply the number parts of all the terms together. Then multiply the variable parts together.
Examples:
a. ( 4x )( -5x ) = ( 4 . -5 )( x . x )
= -20x2
b. (5z2)(3z)(4y)= (5.2.3.4)(y.z.z)
= 120yz2
Only the z is squared
Exploration
Evaluate the following without a calculator:
34 =
33 =
32 =
31 =
Describe a pattern and find the answer for:
30 =
81
27
9
3
1
÷ 3
÷ 3
÷ 3
÷ 3
Zero Power
Anything to the zero
power is oneCan “a” equal zero?
a0 = 1
No.
You can’t divide by 0.
xxxxxxx
7x
3 4x
Exploration
Simplify: x x3 4Use the
definition of exponents to
expand
There are 7 “x” variables
Notice (from the initial
expression) 3+4 is 7!
Product of a Power
If you multiply powers having the same base, add the exponents.
m na
Example
Simplify:
32 9x
2 9 03x x y
113x
1
Add the exponents
since the bases are the same
Anything raised to the 0 power is 1
Practice
Simplify the following expressions:
5
3 0 2
3 5 2 4
1)
2) 2 3
3) 9 4
x x
x z x
x y x y
6x56x
5 936x y
3 3 3 3 3x x x x x
15x
3 5x
Exploration
Simplify: 53x
Use the definition of
exponents to expand
The Product of a Power
Rule says to add all the 3s
Adding 3 five times is equivalent to multiplying
3 by 5. The same exponents from the initial
expression!
Power of a Power
To find a power of a power, multiply the exponents.
m na
1 12 9 22 4s s t t
Example
Simplify: 6 32 3 22 4s s t t
13 118s t
2s 2 6s 3 3t 24t
2 4 1 12s 9 2t
Multiply the powers of a
exponent raised to another
power
Any base without a power, is
assumed to have an
exponent of 1
Multiply numbers without
exponents and add the
exponents when the bases
are the same
Practice
Simplify the following expressions:
42
2 54 3
2 65 4 2
1)
2)
3)
y
a a
x y x x y
8y23a
17 11x y
5x
2 2 2 2 2z x z x z x z x z x
5 10x z
Exploration
Simplify: 52z x
2 5z Use the definition of
exponents to expand
The Product of a Power Rule
says to add the exponents with the same bases
Adding 2 five times is equivalent to
multiplying 2 by 5
Notice: Both the z2 and x were raised to the 5th power!
Power of a Product
If a base has a product, raise each factor to the power
m ma b
20y
20y
2x 4 5y 23
Example
Simplify: 52 43 2x xy
7 20288x y
52 5x
2x9 32 5x2 5x 9 32
Everything inside the
parentheses is raised to the
exponent outside the
parentheses
Multiply numbers without
exponents and add the
exponents when the bases
are the same
Multiply the powers of a
exponent raised to another power
Practice
Simplify the following expressions:
5
4 52 3
32 4
1)
2) 2 2
3) - 2 3
pqr
ab a
x x yz
5 5 5p q r
19 8512a b
7 3 1254x y z
First Four
1. 125x3 9. -15a5b5 16. 64x11
2. 64d6 10. r8s12 17. 256x12
3. a7b7c 11. 36z11 18. 9a8
4. 64m6n6 12. 18x5 19. 729z10
5. 100x2y2 13. 4x9 20. 321
6. -r5s5t5 14. a4b4c6 21. 108a11
7. 27b4 15. 125y1222. -81x17
8. -4x7
Exploration
55 312554 62553 12552 2551 550 15-1
5-2
5-3
5-4
1/32 1/16 1/8 ¼ ½ 1
1
25
1
24
1
23
1
22
1
21
1
20
1
2 1
1
2 2
1
2 3
1
2 4
1/5
1/251/1251/625
2
4
8
16
Complete the tables
(with fractions) by finding
the pattern.
÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
x 2
x 2
x 2
x 2
x 2
x 2
x 2
x 2
x 2
Negative Powers
A simplified expression has no negative exponents.
1ma
ma1ma
Negative Exponents “flip”
and become positive
3b
3
620ba
6 320a b
Example
Simplify: 10 3 44 5a b a
4 5 10 4a All of the old
rules still apply for negative exponents This is not
simplified since there is a negative exponent
Flip ONLY the thing with the
negative exponent to the bottom and
the exponent becomes positive
12
2 3128xy
Example
Simplify:2 1
3
12
8
x y
x
532xy
4
4
2x8
3x1y
Everything with a positive
exponent stays where it is.
Since all of the negative
exponents are gone, apply all of the old rules
to simplify.
Everything with a negative exponent
is flipped and exponent becomes
positive.
Practice
Simplify the following expressions:
3
2
5 3
3 8 4
23
1) 8
62)
4
3) 3 y
4) 2
x
x y
x x
a b
1512
7
332xy
8
7
3y
x
6
24ab
x x x x x x x x x xx x x x x x
10 6x
Exploration
Simplify: 10
6xx
4x
Use the definition of
exponents to expand
Since everything is
multiplied, you can cancel
common factors
The 6 “x”s in the denominator cancel 6
out of the 10 “x”s in the numerator. This is the same as subtracting
the exponents from the initial expression!
Only 4 “x”s remain in the
numerator
Quotient of a Power
To find a quotient of a power, subtract the denominator’s
exponent from the numerator’s exponent if the
bases are the same.
a0
m na
6 2x 26
4 213 x y
Example
Simplify:6
2 3
2
6
x y
x y
4
23xy
1 3y
1
Divide the base
numbers first
Subtract the exponents of
the similar bases since
there is division
Not simplified since there is a
negative exponents
Flip any negative
exponents
Practice
Simplify the following expressions:6 0
3
6
12
9 3
3
1) 5
122)
4
143)
4
a b
a
x
xy
x y
x y
3
5a
5
123xy
6 272x y
a a a a a ab b b b b b a a a a a ab b b b b b
Exploration
Simplify:6
a
b
6
6ab
Use the definition of
exponents to expand
Multiply the fractions
Use the definition of exponents to rewrite.
Notice: Both the numerator and
denominator were raised to the 6th power!
To find a power of a quotient, raise the denominator and numerator to the same power.
Power of a Quotient
m
m
a
b
2y 5 3x 2 6 21
2 15
8
3
y x y
x
Example
Simplify:32 2 7
5
3 2x y
y x
2 6 21
15
8
9
y x y
x
2 3x 7 3y 3223
2 21
15 6
8
9
y
x
23
9
8
9
y
x
Everything in the fraction is raised to the
power out side the parentheses.
Multiply the fractions
Subtract the exponents
when there is division, and
add when there is multiplication
Practice
Simplify the following expressions:
2
83
0
4
2
75
4
1)
22)
3)
a
bc
x
y
s f
zr
24
8ab
8
416
y
x
35
28 14 7
f
r s z