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 = a1∙a1 = a2 Also, b4∙b-3 = b1 = b
(2xy2)3uses 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)
uses a different rule If you are dividing and they have the same base, subtract the exponents. Seem reasonable?
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
Rules that give meaning to sqrts Square root of thirty six: Negative square root of thirty six: Square root of negative thirty six:
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. 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
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. WHEN THE RADICALS ARE FINALLY THE SAME, WE CAN SMASH THEM TOGETHER! (8 OF THEM MINUS 9 OF THEM IS -1 OF THEM.)