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Hisab al-jabr w’al-muqa-balah
This is the title of a 300 year old book (in
Arabic). Translated, it means:
“Science of the reunion and the
opposition”
or
“The science of equations”
Does that part of the title look familiar?
algebra
For algebraic operations, we begin
to mix together numbers and
letters into our operations, which
is a major challenge for students.
By now we know that a variable
represents a quantity that can
change….
A little math magic…..
Think of a number from 1 to 5
Add 3
Multiply by 2
Subtract 4
Divide the number in half
Subtract the number you started
with…
Multiplying and Dividing Powers
Can you think of some examples
of any short- cuts
How about cleaning your room?
Cutting the grass
Dishes? (eat over the sink)
It is our nature to
search for more
efficient ways to do
things
The Exponent Laws are an example of a
mathematical short-cut.
We’ll learn the mechanics of the short cut
first, then we’ll examine the applications.
Specifically, repeated operations can be
compressed using the Exponent Laws
The following examples will illustrate
Through the investigation of
patterns, we are going to derive the
first and second exponent laws….
But first, a few practice runs…
Examine the following
patterns to predict what the
next symbol will be….
O T T F F …..
M T W T F S ….
JFMAM …
Given any pattern, the
simplest progression will
be the implication.
1 3 5 7 9 …
0 2 6 12
…
1 8 27 …
8 4 2 1 …
2 4 8 16
32 …
These examples are called
Sequences
10 X 10 X 10 X 10 X 10 X 10 X 10
We are multiplying 10 by itself 7
times, so this can be rewritten in exponential form as:
power
base
7
10
exponent
23 X 25 =2 X 2 X 2 X 2 X 2 X 2 X 2 X 2
8
= 2
Examine the exponents…
Is there a short cut?
Since the bases can vary, we will use a
variable to represent all cases
In general:
1. Xa X Xb = Xa + b
For example:
(x3) (x8) = x11
(a4) (a3) = a7
(a5) (a) (a3) = a9
(y4) (x2)= x2y4
25  22 = 2 X 2 X 2 X 2 X 2
2X2
= 23
Is there a short cut?
In general:
2. Xa  Xb = Xa - b
For example:
(x7)
(x2)
(a4)
(a3)
= x5
= a1 = a
Evaluate for t = 3 and s = 2
t2 + s3
= (3)2 + (2)3
=3X3+2X2X2
=9+8
= 17
McGraw-Hill
Ryerson
Pg 114 5,6,7
Pg 126 2,4