Question

If 3^x=4^y=12^z prove that z=xy/x+y

Answer 1

Note:

★ a^m × a^n = a^(m + n)

★ a^m / a^n = a^(m - n)

★ a^m = b => a = b^(1/m)

★ a^(1/m) = b^m

★ [ a^m ]^n = a^(mn)

★ a^m × b^m = (a×b)^m

★ a^m / b^m = (a/b)^m

Solution:

Given: 3^x = 4^y = 12^z

To prove: z = xy(x + y)

Proof:

Let ,

3^x = 4^y = 12^z = k

If 3^x = k , then

3 = k^(1/x)

If 4^y = k , then

4 = k^(1/y)

If 12^z = k , then

12 = k^(1/z)

Now,

=> 12 = k^(1/z)

=> 3×4 = k^(1/z)

=> [ k^(1/x) ] × [ k^(1/y) ] = k^(1/z)

=> k^(1/x + 1/y) = k^(1/z)

=> 1/x + 1/y = 1/z

=> (y + x)xy = 1/z

=> 1/z = (x + y)/xy

=> z = xy/(x + y)

Hence proved .