Download Notes: Laws of Exponents

Laws of
Exponents
Tammy Wallace
What is a monomial?
Mono means one.
A monomial is an expression that is a
number, a variable, or the product of a
number and one or more variables.
Examples:
12
𝑦
number
variable
𝑐
−5𝑥 𝑦
3
Product of a number
and variables
2
What about more than one?
Bi means two
Tri means three
Poly means more than one.
A polynomial is a monomial or the sum or difference
of two or more monomials.
REMEMBER: An expression is NOT a polynomial if there
is a variable in the denominator.
Determine whether each expression is a polynomial.
If it is, state what type of polynomial it is.
7𝑦 − 3𝑥 + 4
A trinomial
10𝑥 3 𝑦𝑧 2
A monomial
5
+ 7𝑦
2
2𝑦
NOT a polynomial
because of the variable
in the denominator
Some monomials have exponents.
For example, 𝑥 2 is read as
x to the second power.
X would be the base of the term and 2 would be
the exponent.
State the base and exponent for each monomial.
34
3
The base is _____
4
The exponent is _____
𝑦3
y
The base is _____
3
The exponent is _____
What are exponents?
Exponents means to multiply a number by itself by
on the number of the exponent.
For example, 4³ means to multiply four by itself 4
times.
4 ∙ ______
4 ∙ ______
4 = ______.
64
So, 4³ = ______
Note: This is not the same as 4 ∙ 3 which equals 12.
Expand the following monomials and simplify to a
number if possible.
34
𝑦3
𝟑 ∙ 𝟑 ∙ 𝟑 ∙ 𝟑 = 𝟖𝟏
𝑦∙𝑦∙𝑦
Multiplying Monomials
There are two types of rules to remember when
multiplying monomials.
1. Multiply Rule
2. Power to a Power Rule
Multiplying Rule: When multiplying monomials,
multiply the coefficients and add the exponents.
Keep in mind, a monomial can be a number, a
variable, or the product of numbers and variables.
Multiplying Rule: When multiplying monomials,
multiply the coefficients and add the exponents
• When multiplying monomials, always analyze what is
given and apply the rule based that.
NUMBERS
VARIABLES
(𝑥 2 )( 𝑥 3 )
What part of the rule needs
𝟖
2 (4) = _______ to be applied?
Adding the exponents.
PRODUCT OF NUMBERS AND
VARIABLES
(2𝑎2 )(3𝑎5 )
What part of the rule should be
applied?
Multiplying coefficients AND
adding exponents.
(𝑥 2 )( 𝑥 3 )
𝟔
2 ∙ 3 = ____
𝟐
𝟑
= (𝑥 _______+_______ )
𝟔
= (______𝑎2 )( 𝑎5 )
=
𝟓
𝒙
𝟐
𝟓
𝟔 _________+_________
= (_____𝑎
)
= 𝟔𝒂𝟕
Multiplying Rule: When multiplying monomials,
multiply the coefficients and add the exponents
NUMBERS
VARIABLES
(𝑚𝑛)( 𝑚3 𝑛)(𝑚4 𝑛5 )
𝟕𝟓
𝟐𝟓 ∙ 𝟑= ___
Apply the same rule as above
but only to add exponents to
like variables.
(𝑚𝑛)( 𝑚3 𝑛)(𝑚4 𝑛5 )
𝟑
𝟏
𝟒 _______+______+________
𝟏
𝟏
𝟓
______+_______+_______
𝑚
𝑛
𝟖 𝟕
=𝒎 𝒏
PRODUCT OF NUMBERS AND
VARIABLES
(𝑥 3 )(4𝑥𝑦 5 )
What part of the rule should be
applied? Multiplying coefficients
AND adding exponents.
𝟒
1 ∙ 4 = ____
𝟑
𝟏
𝟓
𝟒
_______+_______
_______
=(_____𝑥
𝑦
)
= 𝟒𝒙𝟒 𝒚𝟓
Multiplying Monomials
When multiplying monomials, the exponents are
added together. However, if any exponent(s) are
located outside the parenthesis, everything inside the
parenthesis has to be raised to that outside power.
NUMBERS
PRODUCT OF NUMBERS AND
VARIABLES
VARIABLES
𝟐
3(5)²
= 3(__)(___)
5 5
= 75
𝒙 𝟒 𝒚𝟑
What exponent is outside the
𝟐
parenthesis? __
inside the
Expand the monomial ______
parenthesis by the exponent
outside
directly _________the
parenthesis.
𝟒
𝟒
𝟑
𝟒 𝟒
𝟑
= (𝑥 _____ )(𝑥 _____ )𝑦 ______
= 𝑥 _____+_______ 𝑦 _____
𝟖 𝟑
=𝒙 𝒚
𝟐 𝟑𝒙𝟒
𝟐
Remember to ONLY expand
inside the
the monomial ___________
parenthesis by the exponent
directly outside
______the parenthesis.
4)
𝟒)(_____𝑥
𝟑 ____
𝟑 ____
= 2(_____𝑥
𝟒
𝟒 )
𝟑 ∙ ___𝑥
𝟑 ____+_____
= 2(____
=2∙
𝟗
𝟖
_____𝑥 ____
𝟖
𝟏𝟖𝒙
= _________
2 4 2
Multiply −7𝑥 𝑦
3𝑥𝑦
= −7𝑥 2 𝑦 4 (−7𝑥 2 𝑦 4 ) 3𝑥𝑦 2
= −7 ∙ −7𝑥 2 𝑦 4 (𝑥 2 𝑦 4 ) 3𝑥𝑦 2
= 49𝑥 2 𝑦 4 (𝑥 2 𝑦 4 ) 3𝑥𝑦 2
= 49 · 3𝑥 2+2+1 𝑦 4+4+2
= 147𝑥 𝑦
5 10
2