Powers of Monomials 4-4
To find a power of a monomial that is already a power, you can use the definition of a power and the rule of exponents for products of powers.
(x5)3 = x5∙ x5∙ x5 = x5+5+5= x15
Notice that (x5)3
= x15 or x5∙3
In general
(am)n = amn
Rule of Exponents for a Power of a Power
For all positive integers m and n
(am)n
= amn
To find a power of a power, you multiply the exponents
(u4)5 = u20
[(-a)2]3 = (a2)3=
a6
To find the power of a product, you can use the definition
of a power AND the commutative and associative properties of multiplication.
(2x)3 = (2x)(2x)(2x) = (2∙2∙2)∙ (x∙
x∙ x)
= 23∙x3
=8x3
Notice
BOTH the 2 and the x are cubed
So (ab)m
= ambm
Rule for
Exponents for a Power of a Product
For
every positive integer m
(ab)m = ambm
To find
a power of a product, you find the power of each factor and then multiply
(-2k)5=(-2)5k5=-32k5
Simplify
(-3x2y5)3
= (-3)3(x2)3(y5)3
= -27x6y15
If a is a nonzero real number and n is a positive integer
a-n = 1/an
so10-3 = 1/103 = 1/1000
( remember this from Math 6 A)
5-4= 1/54 = 1/625
16-1 = 1/16
Let's look at the rule of exponents for division
Rule of Exponents for Division
If a is a non zero real number and m and n are positive integers, then:If m>n
am/an = am-n
If n > m
am/an = am-n
but that means it would be 1/an-m
If m = n
am/an = am-n = a0 = 1
Let's look at the 2nd case using an example
x2/x7 =x2-7 = x-5
but we write that without negative exponents as 1/x5
Looking at the rule above if n > m
And in this case 7> 2
It says that the
simplified form would be 1/x7-2 which is 1/x5
This should help you understand why
a-n = 1/an
recall that for m > n am/an= am-n
More examples
a7/a3 = a7-3= a4
you can also apply this rule when m < n that is when m - n becomes a negative number. For example a3/a7 = a3-7 = a-4
since
a7/a3 and a3/a7 are reciprocals then
a4 and a-4 must also be reciprocals.
Thus
a-4= 1/a4
a5/ a5 = a5-5 = a0
But you already know that a5/a5 = 1
SO, definition of a0
a0 = 1
However, the expression 00 has no meaning
All the rules for positive exponents also hold for zero and negative exponents.
Summary of Rules for Exponents
Let m and n be any integersLet a and b be any non zero integers
We will be reviewing these throughout the year.. but I want you to begin to understand this concept! We will be incorporating it into an upcoming Project!!
Products of Powers
bmbn = bm+n
Example with negative exponents23⋅2-5 = 23+(-5) = 2-2 = 1/22 = 1/4
Quotients of Powers
bm ÷ bn = bm-n
Example with negative exponents63÷67= 63-7= 6-4= 1/64= 1/1296
Power of Powers
(bm)n = bmn
Example with negative exponents(23)-2 = 2-6 = 1/26 = 1/64
Power of a Product
(ab)m= ambm
Example with negative exponents(3x)-2 = 3-2 ⋅x-2 = 1/32⋅1/x2 = 1/9x2
Power of a Quotient
(a/b)m= am/bm
Example with negative exponents(3/5)-2= 3-2/5-2= (1/32)/ (1/52)= 1/32 ÷ 1/52 which means
1/32 ⋅52/1= 52/32= 25/9
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