These SAT-type questions provide review for the multiplication rule of exponents as well as recognizing the need for using the counting principle vs. careful enumeration in an organized list. Both questions need to be given for the effect. Target: PreAlgebra and beyond...
(a) Consider the list 1,2,3,4,5
If a, b and c are assigned different values from the list above, how many different values will result from the expression abc ?
(b) Consider the list 1,2,4,8,16
If a, b and c are assigned different values from the list above, how many different values will result from the expression abc ?
Notes:
(i) To encourage use of exponent rules, do not allow calculator. What variations would make this even more powerful?
(ii) Possible extension: For (a), ask students to make a conjecture regarding the largest possible power, i.e., is it obvious which is the greatest among 512, 415, and 320 ?
Sunday, February 25, 2007
SAT-Type Challenge: Exponents and Combinatorial Thinking
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6 comments:
What kind of curve is x^y = y^x ?
FWIW, list a is easier for finding how many different values there will be, but list b is much easier for finding the greatest possible value.
jonathan--
x^y = y^x, restricted to positive values, is a fascinating graph that would surprise most students! One piece of course is the line y = x, but we know there is at least two other ordered pairs (2,4) and (4,2). Clearly, if (x,y) is one the graph so is (y,x) so the graph is symmetric to y = x. Other than (1,1), neither x nor y could equal 1. I suspect an interesting curve in the first quadrant. Now which piece of software would be best for graphing implicitly defined functions like this! How would students search for a few 'solutions' on their graphing calculators? Start with some given values of y and...
I haven't received any results from the given problems which usually means they aren ot perceived asi interesting (too boring to list! but remember I am writing these for students to review exponents, arithmetic and counting skills.
Comments:
(a) If the base is 2, 3 or 5, does it make sense to use 4C2 = 6 to count the possibilities? Thus, there are 18 possbile values for those bases. If the base is 4, we have duplications, so we only pick up 3 more values, i.e., those in which both factors in the exponent are odd. Throw in 1 more when the base = 1, resulting in 22 different values. Agree?
(b) I found 14, the largest of which was 2^256. Let me know if you agree!
agree on both.
An easier form of my first question is, all quadrants open, 2^x = x^2
For x^y = y^x, the curve in the first quadrant seems to have asymptotes at x=1 and y=1, intersects the line x=y at (e,e), and of course, passes through (2,4) and (4,2).
To determine this, I took the equation y=log(x)/x, which has two solutions for every y between 0 and 1/e (and x between 1 and infinity). These two solutions satisfy x^y=y^x
Interestingly, there is also a point solution at (-2,-4) and (-4,-2). I doubt there are any more such point solutions.
TC
You may want to clarify that it is a^(b*c) and not a^bc . I was using b and c as digits (2^13,2^31 etc.). So I was using permutations (4P2 for bases 2,3,5) instead of combinations.
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