Suppose that $p$ and $q$ are positive numbers for which \[\log_9 p = \log_{12} q = \log_{16} (p + q).\] then $q/p$ can be expressed in the form $(x + \sqrt{y})/z$, where $x$, $y$, and $z$ are positive integers, and $y$ is not divisible by the square of a prime. find $x + y + z$.

Value of (x + y + z) = 8

Step-by-step explanation:

Suppose p and q are the positive numbers for which

from the given expression,

log(p).log(12) = log(q).log(9)

log(q).2log(3) = log(p).log(12) ------(1)

Now

log(q).log(16) = log(p + q).log(12)

2log(4).log(q) = log(p + q).log12 -------(2)

By adding both the equations (1) and (2),

2log(3).log(q) + 2log(4).log(q) = log(12).log(p) + log(12).log(p + q)

log(q)[2log(3) + 2log(4)] = log(12)[logp + log(p + q)]

2log(q).log(12) = log(12).log[p.(p + q)]

2log(q) = log[p.(p+q)]

q² = p(p + q)

Let

a =

a² - a - 1 = 0

from quadratic formula,

a =

a =

a =

If the solution is represented by then  it will be equal to

then x = 1, y = 5 and z = 2.

Now we have to find the value of (x + y + z).

By placing the values of x, y and z,

(x + y + z) = (1 + 5 + 2) = 8

Therefore, value of (x + y + z) = 8

17   x 256=4352

sorry if this is wrong, i'm new.

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