Download - College algebra p2
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P. 2 is about Exponents
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Reminder of what an exponent is.
• Base
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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.
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Simplify (-3ab4)(4ab-3)
• So the actions we took here could be said:______________________________________
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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
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(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.
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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.
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uses a different rule
(kind of like “distributing” the exponent to the top and the bottom)
235
y
x
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uses a different rule
If you are dividing and they have the same base, subtract the exponents. Seem reasonable?
4
7
x
x
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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
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Changing these to positive exponents
x -1 = 23
1
x
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ba
ba2
43
4
12
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223
y
x
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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
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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).
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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
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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
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• 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
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3
3
4
24
75
48
x
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3 6
3 4
40
24
x
a
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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.
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• 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.)
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• Tricky last step
3 43 5416 xx
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Radical versus Exponential Form
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Convert to Exponential Form
53 2
3
3
3
xy
x
x
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• You could see how doing this rewrite could be helpful.
4 32 xx
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Convert to radical form.
2/3
4/3
2/322
2
a
y
yx
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3/1
3/23/4
xy
yx