If 2 to the power x=3 to the power y= 12 to the power z, prove that xy=z(x+2y)?

Question

Use theory of indices.
Please help!!!!!!!!!!!!!!!!!!!!!!!!

Answer 1

2^x = 3^y = 12^z
=>2^x = 3^y = 2^(2z)3^(z)

2^x = 2^(2z)3^z =>
3^(z) = 2^(x-2z) ... 1
2^x = 3^y ...2
from 1 by raising to power y
3^(zy) = 2^(x-2z) y .. 3
from 2
3^(zy) = 2^(x(2z)) ...4
from 3 and 4
(x-2z)y = xz
xy = 2zy+zx = z(x+2y)
QED

Answer 2

x = y = z = 0

Answer 3

to power of what. I get that x=3 and y=12, and xy=36.
if you have 2^x times 3^z, your answer: mutiply the bases and add the indices=> 6^x+z
If you have (2^z)^x then multiply the indices and leave the base as it is 2^(zx)
Hope this can help!:)

Answer 4

2^x = 3^y = 12^z
=>2^x = 3^y = 2^(2z)3^(z)

2^x = 2^(2z)3^z =>
3^(z) = 2^(x-2z) ... 1
2^x = 3^y ...2
from 1 by raising to power y
3^(zy) = 2^(x-2z) y .. 3
from 2
3^(zy) = 2^(x(2z)) ...4
from 3 and 4
(x-2z)y = xz
xy = 2zy+zx = z(x+2y)

Answer 5

given xy=z(x+2y)
hence we can prove that xy=xz+2zy
i.e. xy-xz-2zy=0....dividing throughout by xyz on both sides...we get
1/z-1/y-2/x=0......equation 1
now...let 2^x=3^y=12^z=k
consider 2^x=k....
taking log on both sides we get
x log 2=log k
or x=log k/log2.....
or x=logk to the base 2....(since loga/logb=loga to the base b)
similarly y=logk to the base 3
z=logk to the base 12
going bak to equation 1...putting above x,y,z values we get
1/logk to the base 12-1/logk to the base 3-2/log to the base 2
taking reciprocal of logarithmic terms....
log 12 to the base k-log 3 to the base k-2 log 2 to the base k
log 12 to the base k-(log 3 to the base k+2 log 2 to the base k)
log 12 to the base k-(log 3 to the base k+log 2^2 to the base k)
log 12 to the base k-(log 3 to the base k+log 4 to the base k)
log 12 to the base k-(log (3*4) to the base k)
since bases are equal
log 12 to the base k-log 12 to the base k..
both get cancelled.. hence we get 0....hence proved...

Answer 6

2^x=3^y=12^z u mean? or? not clear.

Answer 7

(x+2y)
[3+2(12)]
(3+24)
27

Answer 8

From what you have written I am struggling to understand what you have to the power of what.

Fancy rewriting it using brackets or something?

Answer 9

write the question better

Answer 10

that doesn't seem right