Hisab al-jabr w’al-muqa-balah

“Science of the reunion and the opposition”

or

“The science of equations”

This is the title of a 300 year old book (in Arabic). Translated, it means:

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….

Think of a number from 1 to 5

Add 3

Multiply by 2

Subtract 4

Divide the number in halfSubtract the number you started

with…

A little math magic…..

Multiplying and Dividing Powers

Can you think of some examples of any short- cuts

It is our nature to search for more efficient ways to

do things

How about cleaning your room?Cutting the grass

Dishes? (eat over the sink)

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 ….

JF M A M …

Given any pattern, the simplest progression will be the implication.

These examples are called Sequences

8 4 2 1 …

1 8 27 …

0 2 6 12 …1 3 5 7 9 …

2 4 8 1632 …

We are multiplying 10 by itself 7 times, so this can be re-written in exponential form as:

10 X 10 X 10 X 10 X 10 X 10 X 10

107power

baseexponent

23 X 25 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2

= 28

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

(a3) =a9(a5) (a)

(y4) (x2)= x2y4

25 22 = 2 X 2 X 2 X 2 X 2 2 X 2

= 23 Is there a short cut?

In general:

2. Xa Xb = Xa - b

For example:

(x7)

(x2)= x5

(a4)

(a3)= a1 = a

Evaluate for t = 3 and s = 2

t2 + s3

= (3)2 + (2)3

= 9 + 8

= 17

= 3 X 3 + 2 X 2 X 2

McGraw-Hill Ryerson

Pg 114 5,6,7

Pg 126 2,4