Hello Vjesus12.VJesus12 wrote:If x and y are positive, what is x+y? $$(1)\ \ 2^x\cdot3^y=72.$$ $$(2)\ \ \ 2^x\cdot2^y=32.$$
The OA is B .
Why is not sufficient the statement (1)? Can any expert help me?
Thanks in advanced.
Let's start with the statement (2): $$2^x\cdot2^y=32\ \Rightarrow\ 2^{x+y}=2^5\ \Rightarrow\ \ \ x+y=5.$$ Hence, this is sufficient.
Now, let's use the statement (1):
If we pick x=3 and y=2 then x+y=5.
But, since x and y not necessary are integers, we can set x=1 and then $$2^x\cdot3^y=72\ \Rightarrow\ \ 2\cdot3^y=72\ \Rightarrow\ \ \ 3^y=36\ \Rightarrow\ \ y\ln\left(3\right)=\ln\left(36\right)\ \Rightarrow\ \ y=\frac{\ln\left(36\right)}{\ln\left(3\right)}.$$ This implies that $$x+y=1+\frac{\ln\left(36\right)}{\ln\left(3\right)}\ne5$$ Therefore, this statement is not sufficient.
This is why the correct answer is B.
