Question

Find the value of $$\dfrac{z\left( x+2y\right) }{xy}$$. If $$2^{x}=3^{y}=12^{z}$$

Answer 1

Hint: In this question it is given that if $$2^{x}=3^{y}=12^{z}$$, then we have to find the value of $$\dfrac{z\left( x+2y\right) }{xy}$$. So to find the solution we have to consider the above equation as k, which will give the value of 2, 3, 12 in the form of k, and by using $12=4\times 3=2^{2}\times 3$ this we will get the value of $$\dfrac{z\left( x+2y\right) }{xy}$$.
Complete step-by-step solution:
Given,
$$2^{x}=3^{y}=12^{z}$$
Let, $$2^{x}=3^{y}=12^{z}=k$$........(1)
Therefore, by taking $1^{st}$ and $4^{th}$ from the above equation, we get,
$$2^{x}=k$$
$$\Rightarrow 2=k^{\dfrac{1}{x} }$$..........(2)
[ if $a^{n}=b$ then $$a=b^{\dfrac{1}{n} }$$]
Similarly, taking $2^{nd}$ and $4^{th}$ part of equation (1), we get,
$$3=k^{\dfrac{1}{y} }$$..................(3)
Now taking $3^{rd}$ and $4^{th}$ part, we can write,
$$12=k^{\dfrac{1}{z} }$$.................(4)

Now ‘12’ can be written as,
$$12=2^{2}\times 3$$
$$\Rightarrow k^{\dfrac{1}{z} }=\left( k^{\dfrac{1}{x} }\right)^{2} \times k^{\dfrac{1}{y} }$$ [ by putting the value of 2, 3, 12 ]
$$\Rightarrow k^{\dfrac{1}{z} }=k^{\left( \dfrac{2}{k} \right) }\times k^{\dfrac{1}{y} }$$ [since, $$\left( a^{m}\right)^{n} =a^{mn}$$]
$$\Rightarrow k^{\dfrac{1}{z} }=k^{\left( \dfrac{2}{x} +\dfrac{1}{y} \right) }$$ [since, $$a^{m}.a^{n}=a^{\left( m+n\right) }$$]
Now as we know that if $a^{m}=a^{n}$, then $m=n$, so by this we can write the above equation as,
$$k^{\dfrac{1}{z} }=k^{\left( \dfrac{2}{x} +\dfrac{1}{y} \right) }$$
$$\Rightarrow \dfrac{1}{z} =\dfrac{2}{x} +\dfrac{1}{y}$$
$$\Rightarrow \dfrac{1}{z} =\dfrac{2y+x}{xy}$$
$$\Rightarrow \dfrac{2y+x}{xy} =\dfrac{1}{z}$$
$$\Rightarrow \dfrac{z\left( 2y+x\right) }{xy} =1$$ [multiplying both side by ‘z’]

Therefore we get, $$\dfrac{z\left( 2y+x\right) }{xy} =1$$
Which is our required solution.
Note: To solve this type of question you have to keep in mind that whenever you have given this type of equation($$2^{x}=3^{y}=12^{z}$$) where more than 1 equations are included, then you have to consider each part of given equation as k(constant value), this will help you to find the solution in a simple way. Also in order to get the solution, try to find a link in between the obtained equations, like I have used $12=2^{2}\times 3$ to connect the equation (2),(3) and (4).