special tricks for these?

[gemini9525]

or do you just have to do them straight out?

(21^3)*(15^2)/(9^4)
*straight out would be to divide each () by 9...similar problems have occured on all of the uil tests this year...wondering if there's a faster way*

(3-5i)(3-5i)=a+bi, find a+b
*foil? that takes forever...*

the sum of the first eleven terms of the Fibonacci sequence 2,4,6,10,16,26...
*my trick is 11*(8th term)+a. but on uil when they only give you first 6, it's takes a little long to add out the 8th term, then do the trick. i heard there are ways to manipulate just the first two terms. would anyone nice offer a hint? also for the sums of the first 8,9,10 terms?*

[bradp]

1) 21^3 * 15^2 / 9^4 broken down into prime factors = 7^3 * 5^2 * 3^5 / 3^8

threes cancel so you're left with 343 * 25 / 27
25/27 is close to 1 so subtract a bit from 343.... ~320

or....

21^3 = ~9200 divided by 9^4 = ~6500 is a bit less than 1.5.. multiply that by 225.
I wouldn't do that though.

2) foil.. 9 + 25i^2 = -16, -15 + -15 = -30 --> -46

3) take the two starting terms as x and y - the sequence is then x + y + (x+y) + (x+2y)... add em all up and you get formulas for the sums. The sum of the first 11, 89x + 143y, has fairly large numbers... and it's debatable whether it's faster or slower than 11f(8) + x

[AuSmith]

2) Set i=1.

3) If I remember correctly, they handpicked 11 terms because 11 is a Lucas number. So, 8,9,10 terms doesn't seem to fit. You could of course do similar things to 89x+144y, but that's lame. I'd rather just lose.

[suchenzang]

aaron, i don't think this is like summing the coefficients of (ax+by)^n by adding a and b and then raising it to the nth power. i=1 will give you 4, unless i'm reading your meaning behind your solution to number 2 incorrectly...

[AuSmith]

Thank you, Susan. You deftly parried my depthless humor. Ha, yea, I wasn't serious. Sorry.

[suchenzang]

ouch. i totally didn't catch that. my sense of humor must've died recently. =(

[315937]

[AuSmith]

Whoah, I had assumed this was a recent post. Then I saw that I had already replied!

[armesty]

that number guy is annoying, he keeps bringing back old posts and reposting solutions