P. 2 is about Exponents

Reminder of what an exponent is.

• Base

A bunch of rules!

• In the textbook’s P.2, there are a bunch of rules listed. (Some are on or near page 12.)

• To save class time, I will present these rules simultaneous with example problems so you see them in practice, which is the most important part.

• If you want the list of rules to copy in your notes, see the book.

Simplify (-3ab4)(4ab-3)

• So the actions we took here could be said:______________________________________

What was the exponent rule used?

When you multiply things with the SAME BASE, you add the exponents.

For example, a a = a∙ 1 a∙ 1 = a2

Also, b4 b∙ -3 = b1 = b

(2xy2)3 uses a different rule:

• 1st step is like “distributing” the exponent.= (2)3 (x)3 (y2)3 draw little arrows

• 2nd step uses a rule that when you take a power of a power, you multiply the exponents. [so (y2)3 would become y6]

= 8x3y6 is the final answer.

3a(-4a2)0 uses another rule.

• A rule says that ANYTHING to the zero power is ONE.

= 3a 1∙= 3a

• Be careful not to jump to conclusions and automatically put down ONE as the answer to the entire problem: common mistake made.

uses a different rule

(kind of like “distributing” the exponent to the top and the bottom)

235

y

x

uses a different rule

If you are dividing and they have the same base, subtract the exponents. Seem reasonable?

4

7

x

x

rule giving meaning to negative exponents

• If you ever have a negative exponent in your answer, you will need to change it to a positive by crossing it over the magic division bar.

• Change to

• Change to

3

2 7ab

25

4y

x

Changing these to positive exponents

x -1 = 23

1

x

ba

ba2

43

4

12

223

y

x

Rules that give meaning to sqrts

Square root of thirty six:

Negative square root of thirty six:

Square root of negative thirty six:

36

36

36

Exponents versus Indexes

• When an exponent is not written, it is understood to be a one (like raising something to the first power)

• When an index is not written on a radical, it is understood to be a two (as in a square root).

When a root is not a square root:

We used the “break-it-down” rule AND we had to know the meaning of that little three.

3

64

125

I know you can’t use calculators, but you’ll recognize some of these perfect squares and perfect cubes after some practice:3x3x3=27 4x4x4=64 5x5x5=125

When are square roots not even REAL?

• The odd root of a negative will be a negative real:

• The even root of a negative is “No Real Number.”

5 32

4 81

• On page 16ish, there is a list of properties pertaining to radicals. Here are a few of them in action:

3 3

33 5

28

x

3

3

4

24

75

48

x

3 6

3 4

40

24

x

a

Adding/Subtracting Radicals• In order to add or subtract radicals, they

MUST be “similar” first. (The radical parts need to look the same.)

• To me, this is reminiscent of how fractions must have similar denominators in order to be added or subtracted.

• In order to try to make them similar, all you can do is try to simplify each radical.

• Simplify all radicals first.

273482

WHEN THE RADICALS ARE FINALLY THE SAME, WE CAN SMASH THEM TOGETHER! (8 OF THEM MINUS 9 OF THEM IS -1 OF THEM.)

• Tricky last step

3 43 5416 xx

Radical versus Exponential Form

Convert to Exponential Form

53 2

3

3

3

xy

x

x

• You could see how doing this rewrite could be helpful.

4 32 xx

Convert to radical form.

2/3

4/3

2/322

2

a

y

yx

3/1

3/23/4

xy

yx