Algebra Honors (Periods 6 & 7)

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 x^5∙3

In general

(a^m)ⁿ = a^mn

Rule of Exponents for a Power of a Power

For all positive integers m and n

(a^m)ⁿ = a^mn

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 = a^mb^m

Rule for Exponents for a Power of a Product

For every positive integer m

(ab)^m = a^mb^m

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

Negative Exponents 
If a is a nonzero real number and n is a positive integer

a^-n = 1/aⁿ

so
 10^-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

a^m/aⁿ = a^m-n 

If n > m

a^m/aⁿ = a^m-n  

but that means it would be 1/a^n-m

If m = n

a^m/aⁿ = a^m-n  = a⁰ = 1

Let's look at the 2nd case using an example

x²/x⁷ =x^2-7 = 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/x^7-2 which is 1/x⁵

 This should  help you understand why
a^-n = 1/aⁿ
recall that for m > n a^m/aⁿ= a^m-n
More examples
a⁷/a³ = a^7-3= a⁴
you can also apply this rule when m < n that is when m - n becomes a negative number. For example a³/a⁷ = a^3-7 = 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 = a^5-5 = 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 integers
Let 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 

b^mbⁿ = b^m+n

Example with negative exponents
2³⋅2^-5 = 2^3+(-5) = 2^-2 = 1/2² = 1/4

Quotients of Powers 

b^m ÷ bⁿ = b^m-n

Example with negative exponents
6³÷6⁷= 6^3-7= 6^-4= 1/6⁴= 1/1296

Power of Powers

(b^m)ⁿ = b^mn

Example with negative exponents
(2³)^-2 = 2^-6 = 1/2⁶ = 1/64

Power of  a Product

(ab)^m= a^mb^m

Example with negative exponents
(3x)^-2 = 3^-2 ⋅x^-2 = 1/3²⋅1/x² = 1/9x²

Power of  a Quotient

(a/b)^m= a^m/b^m

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