how to simplify?

Question

If r^2/4(3x)^r(2/9x^2)^6-r can be simplified to k/x^3, find the values of the constants r and k.

Answer 1

I think you meant to type r^2/4 * (3x)^r * (2/(9x^2))^(6-r)...
... in which case, I've got that r =3 and k = 2/3...

I did it out the way you typed it first and got some crazy stuff, when I reworked it, this way is much easier! Let me know what you meant. :-) Having the 9x^2 in the entire denominator is an important assumption, as is the 6 - r power

If I interpreted this correctly I've got something like this...
r^2/4 * (3x)^r * (2 / (9x^2) )^(6-r) = k/x^3

Clean up the powers first...
On the top... r^2 * 3^r * x^r * 2^(6-r)
On the bottom... 4 * 9^(6-r) * (x^2)^(6-r)

Use some rules of exponents...
base of r... r^2... no pair

base of 3... 3^r / 9^(6-2r) = 3^r / (3^2)^(6-2r) = = 3^r / 3^(2(6-2r)) = 3^r / 3^(12-2r) = 3^(r-(12-2r)) = 3^(r-12+2r) = 3^(3r-12)

base of 2... 2^(6-r) / 4 = 2^(6-r) / 2^2 = 2^(6-r-2) = 2^(4-r)

base of x... x^r / (x^2)^(6-r) = x^r / x^(2*(6-r)) = x^r / x^(12-2r) = x^(r-(12-2r))= x^(r-12+2r) = x^(3r-12)

These are all factors, so we've got ACE OF BASE! HA!
r^2 * 3^(3r-12) * 2^(4-r) * x^(3r-12) = k*x^(-3)

I'm acutually concerned about the exponent on the x... it has to be -3 on both sides... so
3r - 12 = -3
3r = 9
r = 3

Now for k... it's the stuff on the front end of the left hand side. So let's plug in 3 for r and clean it up...
k = r^2 * 3^(3r-12) * 2^(4-r)
k = 3^2 * 3^(3*3-12) * 2^(4-3)
k = 9 * 3^(9-12) * 2^1
k = 9 * 3^(-3) * 2
k = 9 * 1/27 * 2
k = 18 / 27
k = 2/3

Answer 2

r^2/4(3x)^r(2/9x^2)^6-r = k/x^3
simplifying the LHS,
= (r^2 / 4) (9 x^2)^(r/2) (2 / (9 x^2))^(6-r)
= (r^2) (2)^(6-r-2) (9 x^2)^(r/2 - 6 +r)
= (r^2) (2)^(4-r) (9 x^2)^(3r/2 - 6) = RHS = k(x)^-3
Observe the terms in x, (specifically the x raised to the powers of)
(x^2)^(3r/2 - 6) = (x)^-3
hence,
2(3r/2 - 6) = -3
r=3

do the same to the coefficients of x,
(r^2) (2)^(4-r) (9)^(3r/2 - 6) = k
sub in r=3, to obtain:
k=2/3

cheers! (:
i suggest u write it down on paper first, it's pretty messy to read it off the screen.

Answer 3

If you are an intelect trying to show off, good luck. Try getting a boy or girl friend. If you're serious, I'm lost.