multiplication properties of exponents and...multiplication properties of exponents monomial a...
TRANSCRIPT
Multiplication Properties of Exponents
Monomial A number, variable, or the product of a number and
one or more variables.
Constant Any monomial that is a real number.
Expression Monomial ? Reason
-5 yes Constant, real number
p + q no The plus sign
x Yes variable
๐
๐ no variable denominator
๐๐๐8
5 yes =
1
5๐๐๐2
โข Exponents or Powers
Be careful with negative bases:
โ(๐)๐ = โ๐ โ ๐ โ ๐ โ ๐ โ ๐ = โ๐๐ (โ๐)๐ = โ๐ โ โ๐ โ โ๐ โ โ๐ = ๐๐
EXPONENT RULES GRAPHIC ORGANIZER
Name Rule Examples
Adding & Subtracting Monomials
COMBINE LIKE TERMS!!
(Do Not Change common variables and exponents!)
1. 9๐ฅ2๐ฆ โ 10๐ฅ2๐ฆ = โ1๐ฅ2๐ฆ
2. Subtract 6w from 8w 8w-6w=2w
Product Rule
๐ฅ๐ โ ๐ฅ๐ = ๐ฅ๐+๐
Keep the base, add exponents
1. โ2 โ โ6 = โ8
2. (โ2๐2๐) โ (7๐3๐) =
โ14๐5๐2
Power Rule
(๐ฅ๐)๐ = ๐ฅ๐โ๐
Keep the base, multiply exponents
1. (๐ฅ2)3 = ๐ฅ6
2. (โ2๐5)2 โ ๐3 =
4๐13
Quotient Rule
๐ฅ๐
๐ฅ๐= ๐ฅ๐โ๐
Keep the base subtract exponents
1. 27๐ฅ5
42๐ฅ=
27๐ฅ4
42
2. (๐ฆ2)2
๐ฆ4 =๐ฆ4
๐ฆ4 = 1
Negative
Exponent Rule
๐ฅโ๐ =1
๐ฅ๐
1
๐ฅโ๐= ๐ฅ๐
1. โ5๐ฅโ2 =โ5
๐ฅ2
2. 4๐2
8๐5 =1
2๐3
Zero Exponent
Rule
๐ฅ0 = 1
1. 7๐ฅ0 = 7(1) = 7
2. (๐ค4)2
๐ค8 =๐ค8
๐ค8 = 1
You Try:
Letโs practice the Product Rule, Power Rule and Zero Exponent Rule!
Simplify x3 โ x2 โ x6 = x3+2+6
= x11
Simplify 4x4y7 โ 2xy5 โ x6
= (4)(2)(x4+1+6)(y7+5) = 8x11y12
Simplify (a7b8)5 = a(7)(5)b(8)(5)
= a35b40
Simplify (3x2y6z3)4
= 34x(2)(4)y(6)(4)z(3)(4)
= 81x8y24z12
Simplify (2m3n2)2 โ (m2n2)3 = 22m(3)(2)n(2)(2) โ m(2)(3)n(2)(3)
= 4m6n4 โ m6n6 = 4m6+6n4+6 = 4m12n10
Simplify 8m5n4 โ (2mn3)2
= 8m5n4 โ 22m(1)(2)n(3)(2)
= 8m5n4 โ 4m2n6 = 32m5+2n4+6 = 32m7n10
Simplify 3x0y3z6 โ 7y0z2 = 3(1)y3z6 โ 7(1)z2
= (3)(7)y3z6+2 = 21y3z8
Simplify (9a7b3c12)0
= 1
Division Properties of Exponents
Quotient of Powers
When dividing monomials
with the same base, We subtract the
Exponents Divide or simplify coefficients!
๐ฅ5
๐ฅ3= ๐ฅ5โ3 = ๐ฅ2
------------------------- 8๐ฆ9
2๐ฆ3= 4๐ฆ6
Power of a Quotient
Find the power of the numerator and the
power of the denominator, then
subtract.
(๐ฅ
๐ฆ)3 =
๐ฅ3
๐ฆ3
--------------------------
(3๐ฅ2
4๐ฅ)2 =
9๐ฅ4
16๐ฅ2=
9
16๐ฅ2
For more complicated division problems line up like variables before applying
the rules:
6๐ฅ5๐ฆ3๐ง2
2๐ฅ๐ฆ๐ง
= 3๐ฅ4๐ฆ2๐ง
3๐4๐2
6๐3
=1๐๐2
2
Negative Exponents: We need to โfixโ any nonzero number raised to a negative
exponent.
When solving problems with negative exponents, circle the variable with the
negative exponent and move just that variable and negative exponent to the
other side of the fraction bar and the exponent becomes positive.
NOTE: NEGATIVE COEFFICIENTS DONโT GET MOVED UNLESS THEY ALSO
HAVE A NEGATIVE EXPONENT!!
โ2๐โ1๐2
๐โ3= โ
2๐2๐3
๐
You Try!
Simplify 7๐6๐4๐2๐5
21๐4๐2
= 1๐2๐4๐2๐3
3
Simplify 24๐12๐6
8๐8๐
= 3๐4๐5
Simplify ( 5๐ฅ3
15๐ฅ2)2
= 25๐ฅ6
225๐ฅ4
= ๐ฅ2
9
Simplify ๐7๐3
๐4๐2๐โ3
= ๐3๐๐3
Simplify 15๐ฅ5๐ฆโ2
๐งโ4
= 15๐ฅ5๐ง4
๐ฆ2
Simplify ๐7๐2๐4
๐3๐5๐4
= ๐4๐โ3๐0
= ๐4
๐3
Rational Exponents
Radical Sign When there is no number in the elbow, we assume โ2โ
โ25 = 5*5
โ83
is read cube root of 8 and is equal to 2 * 2* 2
Exponential expression โ Rational Expression
62
3 โ (โ63
)2 The denominator of the fraction is the
elbow of the radical and the numerator is the exponent.
Letโs go from exponential to rational and rational to exponential:
5๐ฅ1
2 = 5โ๐ฅ
(5๐ฅ)1
2=โ5๐ฅ
Solve Exponential Equations: In an exponential equation, the variable is an
exponent. To solve, we must make sure both sides of the equal sign have the
same base โ then we can set the exponents equal to each other.
If 5x = 57 then x = 7. If 2x = 8 then 2x = 23 and x = 3
You Try:
6๐ฅ = 216
6๐ฅ = 63
๐ฅ = 3
82x = 16
(23)2x = 24
6x = 4 Power to a power
x = 4
6=
2
3
25๐ฅโ1 = 5
(52)๐ฅโ1 = 51 Power to a power
52(๐ฅโ1) = 51
2(๐ฅโ1) = 1
2๐ฅโ2 = 1
2๐ฅ = 3
๐ฅ =
27x-5 = 9
27x-5 = 32
x โ 5 = 2
x = 7
1
9
Exponential Functions
Exponential Functions Non-linear
๐ฆ = ๐๐๐ฅ ๐คโ๐๐๐ ๐ โ 0 ๐๐๐ ๐ โ 1
โข The exponent is the variable
๐ฆ = ๐๐๐ฅ + ๐ (c value indicates range)
Graph ๐ฆ = 3๐ฅ
x y
-2 (๐ฆ = 3โ2)
-1 1
3
0 1
1 3
2 9
y-int=1 domain = all real numbers range = {y ฤฎ y>0} c = 0
Exponential Growth Exponential Decay
๐ = ๐๐๐ ๐ = ๐๐๐ Growth Decay
a > 0, b > 0 a > 0, 0 < b < 1
(b is a decimal or fraction)
Domain: All real numbers Domain: All real numbers
Range: Positive Real Numbers Range: Positive real #s.
Identifying Exponential Behavior:
Method 1 Method 2
Domain (x) regular intervals. (+5)
Range (y) common factor (1/2)
YES! Exponential Yes! Exponential Graph!
x 0 5 10 15 20 25 y 64 32 16 8 4 2
Graph ๐ฆ = (1
4)๐ฅ . Find the y-intercept and domain and range.
x y -2 16 -1 4 0 1 1 1
4
2 1
16
y-int = 1
domain= all real #
range = {y ฤฎ y>0}
Graph ๐ฆ = 2(3)๐ฅ + 1. Find the y-intercept and domain and range.
x y -2 1
2
9
-1 12
3
0 3 1 7 2 19
y-int = 3
domain= all real #
range = {y ฤฎ y>1}
Growth and Decay
๐ฆ =๐ถ(1+๐)๐ก
๐ฆ =๐ถ(1โ๐)๐ก
๐ด =๐(1+๐
๐)nt
Y = final amount
C = Initial Amount r
= rate of change
(as decimal) t = time
Y = final amount
C = Initial Amount r
= rate of change
(as decimal) t = time
A = current amount
P = Principal - Initial Amount
r = rate of interest (as decimal) t =
Number of years
n= Number of times compounded
Exponential
Growth
Exponential
Decay
Compound
Interest
Examples:
1. The prize for a radio station contest begins with $100 gift card. Once a day a
name is announced. The person has 15 minutes to call or the prize increases by
2.5% for the next day.
a. Write an equation to represent the amount of the gift card in dollars after t
days with no winners.
b. How much will the gift card be worth if no one wins after 10 days?
a. ๐ฆ = ๐(1 + ๐)๐ก (since the amount grows, we use growth
formula)
๐ฆ = 100(1 + .025)๐ก (plug in what you know. C=100; r โ 2.5% =
.025)
๐ฆ = 100(1.025)๐ก equation that represents amount of the gift
card in dollars after t days with no winners.
b. ๐ฆ = 100(1.025)๐ก (Substitute 10 for t and solve.)
๐ฆ โ 128.01
In 10 days, the gift card will be worth $128.01
2. Mariaโs parents invested $14,000 at 6% per year compounded monthly. How much
money will be in the account after 10 years?
๐ด = ๐(1 + ๐
๐)๐๐ก
What do we know: P = 14,000
r = 6% or .06
n = 12 (monthly) t = 10
๐ด = 14000(1.005)120
๐ด
๐ด โ 25471.55
There will be about $25,471.55 in the account after 10 years.
Geometric Sequences as Exponential Functions
Geometric Sequence A sequence in which each term after the nonzero
first term is found by multiplying the previous
term by a constant called the common ratio (r). r
โ 0 ๐๐ 1.
Remember
When we looked at arithmetic sequence, we looked for a common difference
between the terms.
Well
For geometric sequence we look for a common ratio. (HINT: we can do this by
dividing the last term by the one before it and repeat)
a) 1, 4, 16, 64, 256 โฆ Common Ratio = 4, yes this is geometric
sequence
b) 1, 3, 5, 7, 9 โฆ.
This is NOT a geometric sequence because
it has a common difference not
a common ratio! Therefore it is
arithmetic sequence
c) 4, 9, 12, 18โฆ
This is neither a geometric sequence nor an
arithmetic sequence.
To find the next few terms of a geometric sequence:
Step 1: Find the common ratio
Step 2: Multiply the last term by the common ratio to get the next term
Find the next three terms in the geometric sequence:
20, -28, 39.2, _____, _____, _____
Common ratio = = โ1.4
Next three terms: (39.2)(-1.4) = -54.88
(-54.88)(-1.4) = 76.832
(76.832)(-1.4) = -107.5648
nth term: just like we were able to find the nth term of the arithmetic
sequence using a formula, we can find the nth term of a
geometric sequence:
๐๐ = ๐๐ โ ๐๐โ๐ where ๐1= first term in sequence
a. Find the ninth term for the sequence -6, 12, -24, 48
๐1 = โ6 ๐ = 9 ๐ = โ2
๐๐ = โ๐ โ (โ๐)๐โ๐
= โ๐(โ๐)๐
= โ๐๐๐๐
b. Write an equation for the nth term:
๐๐ = โ๐(โ๐)๐โ๐
Recursive Formulas
Recursive Formula Allows you to find the nth term of a sequence
by performing operations to one or more of
the preceding terms.
USING RECURSIVE FORMULAS:
Substitute the first term into the given equation to find the 2nd term.
Substitute the 2nd term into the given equation for ๐๐โ1 to find the 3rd term.
Repeat until youโve found all the terms asked for:
Find the first five terms of the sequence in which:
๐1 = 7 ๐๐ = 3๐๐โ1 โ 12 if nโฅ 2
First term given equation
๐1 = 7 given
๐2 = 3(7) โ 12 = 9
๐3 = 3(9) โ 12 = 15
๐4 = 3(15) โ 12 = 33
๐5 = 3(33) โ 12 = 87
Sequence: 7, 9, 15, 33, 87โฆ
WRITING A RECURSIVE FORMULA:
Step 1: Step 2: Step 3:
Arithmetic
17, 13, 9, 5, โฆ
Common difference = -4
๐๐ = ๐๐โ1 + ๐
๐๐ = ๐๐โ1 + (โ4)
๐๐ = ๐๐โ1 โ 4
๐1 = 17 ๐๐ = ๐๐โ1 โ 4 ๐ โฅ 2
Geometric
6, 24, 96, 384 โฆ
Common Ratio = 4
๐๐ = ๐ โ ๐๐โ1
๐๐ = 4 โ ๐๐โ1
๐1 = 6 ๐๐ = 4๐๐โ1 ๐ โฅ 2
Step 1: Determine if sequence is arithmetic or geometric โ find
common difference or common ratio.
Step 2: Arithmetic โ ๐๐ = ๐๐โ1 + ๐
Geometric โ ๐๐ = ๐ โ ๐๐โ1
Step 3: State the first term and domain for n