How can I solve dimensional analysis problems involving exponents and constants?

[nilesthebrave]

Hi, sorry for asking this but my brain still seems to be on lockdown from the summer. I have a pretty good idea of how dimensional analysis works and only seem to be having issues on one type of problem currently. something like:

t=(Cm^x)(k^y)

where:
t-oscillations of mass
m-mass
spring constant-k(force/length)
C-dimensionless constant

to find x and y.

T=(M^x)(Force/L)^y=(M^x)(ML/L(T^2))^y
T=(M^x)(M/T^2)^y
T=(M^x)(M^y/T^2y)
T=(M^(x+y))(M^y/T^2y)
T(T^2y)=M^(x+y)
T^(2y+1)=M^(x+y)

Then you get

2y+1=0
x+y=0

solving for y at top equation:
y=-1/2
then plugging in for second equation you get
x=1/2

So I have that one.

Now where I'm having hangups is on one like say:

v=(CB^x)(p^y)

B-bulk modulus
p-density
c-dimensionless constant
v-velocity
Find x and y

So I know I start with:

L/T=(M/LT^2)^x(M/L^3)^y

But honestly, I get stuck at this point. I can't figure out how to get things to cancel or how to make things simplify down easier. Do I distribute the exponent? Do I multiply the left hand by a reciprocal of one of those? Honestly, I don't get how to do one like this even though I fully get the first one which is fairly similar. Any hints to nudge me in the right direction to solve this, its been bugging me for awhile now.

 

[Pi-Bond]

Simplify the exponents and compare them on both sides, just like you did in the last question (for example the exponent of L must be equal on both sides.)