Question

TOPIC: Indices and logarithms.
Simplify 81^((n)/(4) + 1) - 3^(n-1)

Answer 1

the answer that my teacher gave me is 242(3^n-1). I still couldn't get the real steps to the answer.

Answer 2

Eureka! Got it!
ok, so we have this
81^((n/4)+1) - 3^(n-1)=
81^((n/4)+1) - 81^((1/4)(n-1))=
81^(n/4)*81^1 - 81^(n/4)*81^(-1/4)
81^(n/4) * (81 - 81^(-1/4))
3^n * ( 81 - 1/3 )
3^(n-1) * 3 * (81-1/3)
3^(n-1) * (243-1)
3^(n-1) * 242

Answer 3

I skipped a few steps, so if anything seems unclear, don't hesitate to ask

Answer 4

okay, let me copy this steps first, hehe

Answer 5

can you show me the steps that you skipped or explain it, I'm a bit confused on the step 5.

Answer 6

right, no problem, so let's start at 4)
81^(n/4) * (81 - 81^(-1/4))
->(81^(1/4))^n * (81 - 81^(-1/4))
->81^(1/4) = 3 so
3^n * (81 - 81^(-1/4))

Answer 7

i guess i'm too slow, i'm still trying to understand it.

Answer 8

Are you in a hurry? I'm at work right now, so I don't have much time to spare for the moment. Do you mind if I go through the steps a bit later?

Answer 9

i don't mind

Answer 10

I got it! I finally GOT it!! Thanks :)

Answer 11

Alright, so I'm basically only using 2 things here :
Distributivity and the rule a^(x+y) = a^x*a^y
first thing I did was to get a common factor in both terms, 81^(n/4). Since 81^(1/4) = 3, it's easy to substitute the 3 in the second term, which gives us this (I'll start over the steps I guess, and I've deleted I had written that was wrong)
81^(n/4 + 1) - 3^(n-1)
81^(n/4 + 1) - (81^(1/4))^(n-1) (Direct substitution, no other operations in this step)
so, given the rule (a^n)^m = a^(nm) we can now write this
81^(n/4 + 1) - (81^((1/4)(n-1))
Distributivity on the second exponant
81^(n/4 + 1) - (81^(n/4-1/4)
now that's where we use a^(m+n) = a^m*a^n
81^(n/4) * 81^1 for the first term
81^(n/4) * 81^(-1/4) for the second
new equation is
81^(n/4)*81 - 81^(n/4) * 1/3 (81^(-1/4) = 1/3)
now, we can factorize 81^(n/4) since it multiplies both terms
81^(n/4)(81 - 1/3)
Again, we use a^(mn) = (a^n)^m
(81^(1/4))^n * (242/3)
81^(1/4) = 3, as previously stated, so we can write
3^n * (242/3)
This is where, if you hadn't given me the answer, I would've stopped, because the next step is not that obvious (at least, not for me) but here it is.
3^n * 1/3 * 242
->1/3 = 3^(-1)
3^n * 3^(-1) * 242
Once again, a^n * a^m = a^(n+m)
3^(n-1) * 242

Answer 12

i wasn't about to delete that wall of text I was writing ;-)
Huzzah on getting it, it wasn't simple and a very good question indeed!

Answer 13

Right!