Question
If $2^\text{x}=3^\text{y}=12^\text{z},$ show that $\frac{1}{\text{z}}=\frac{1}{\text{y}}+\frac{2}{\text{x}}.$
✓
Answer
Let $2^\text{x}=3^\text{y}=12^\text{z}=\text{k}$ Then, $2=\text{k}^{\frac{1}{\text{x}}},\ \text{3}=\text{k}^{\frac{1}{\text{y}}}$ and $12=\text{k}^{\frac{1}{\text{z}}}$ Now,$12=\text{k}^{\frac{1}{\text{z}}}$
$\Rightarrow2^2\times3=\text{k}^{\frac{1}{\text{z}}}$
$\Rightarrow2^2\times3=\text{k}^{\frac{1}{\text{z}}}$
$\Rightarrow\Big(\text{k}^{\frac{1}{\text{x}}}\Big)^2\times\text{k}^\frac{1}{\text{y}}=\text{k}^{\frac{1}{\text{z}}}$
$\Rightarrow\text{k}^\frac{2}{\text{x}}\times\text{k}^\frac{1}{\text{y}}=\text{k}^\frac{1}{\text{z}}$
$\Rightarrow\text{k}^{\frac{2}{\text{x}}+\frac{1}{\text{y}}}=\text{k}^\frac{1}{\text{z}}$
$\Rightarrow\frac{2}{\text{x}}+\frac{1}{\text{y}}=\frac{1}{\text{z}}$
$\Rightarrow2^2\times3=\text{k}^{\frac{1}{\text{z}}}$
$\Rightarrow2^2\times3=\text{k}^{\frac{1}{\text{z}}}$
$\Rightarrow\Big(\text{k}^{\frac{1}{\text{x}}}\Big)^2\times\text{k}^\frac{1}{\text{y}}=\text{k}^{\frac{1}{\text{z}}}$
$\Rightarrow\text{k}^\frac{2}{\text{x}}\times\text{k}^\frac{1}{\text{y}}=\text{k}^\frac{1}{\text{z}}$
$\Rightarrow\text{k}^{\frac{2}{\text{x}}+\frac{1}{\text{y}}}=\text{k}^\frac{1}{\text{z}}$
$\Rightarrow\frac{2}{\text{x}}+\frac{1}{\text{y}}=\frac{1}{\text{z}}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Factorize the following expressions:
$(x + 2)^3 + (x - 2)^3$
→$(x + 2)^3 + (x - 2)^3$
If O is the centre of the circle, find the value of x in the following figures:
If $x=2$ is a root of the polynomial $f(x)=2 x^2-3 x+7 a$, Find the value of $a$.
→i. Can we draw a line parallel to the X-axis at a distance of 6 unIts from It and below the X-axis?
ii. Will all of the points $(-3,-6),(10,-6),\left(\frac{1}{2},-6\right)$ be on that line?
iii. What would be the equation of this line?
→ii. Will all of the points $(-3,-6),(10,-6),\left(\frac{1}{2},-6\right)$ be on that line?
iii. What would be the equation of this line?
Solve the following sets of simultaneous equations : 2x + y = 5 ; 3x – y = 5
A 4cm cube is cut into $1cm$ cubes. Calculate the total surface area of all the small cubes.
→Calculate the value of x in the following figures.
In figure, ABCD, ABFE and CDEF are parallelograms. Prove that $\text{ar}(\triangle\text{ADE})=\text{ar}(\triangle\text{BCF}).$
→Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.
Factorize the following expressions:
$a^3 + 3a^2b + 3ab^2 + b^3 - 8$
→$a^3 + 3a^2b + 3ab^2 + b^3 - 8$