Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# If $x$ is a positive integer satisfying ${x^7} = k$ and ${x^9} = m$, which of the following must be equal to ${x^{11}}$?A. $\dfrac{{{m^2}}}{k}$B. ${m^2} - k$C. ${m^2} - 7$D. $2k - \dfrac{m}{3}$E. $k + 4$

Last updated date: 11th Aug 2024
Total views: 429k
Views today: 7.29k
Verified
429k+ views
Hint: In this question, we will proceed by writing the given data and converting ${x^{11}}$ in terms of ${x^7}$ and ${x^9}$. Then substitute the values of ${x^7}$ and ${x^{11}}$ in the converted form of ${x^{11}}$ to get the required answer. So, use this concept to reach the solution of the given problem.

Given that ${x^7} = k$ and ${x^9} = m$.
We have to find the value of ${x^{11}}$.
Now, consider the value of ${x^{18}}$
$\Rightarrow {x^{18}} = {\left( {{x^9}} \right)^2} \\ \Rightarrow {x^{18}} = {\left( m \right)^2}{\text{ }}\left[ {\because {x^9} = m} \right] \\ \therefore {x^{18}} = {m^2} \\$
We can write ${x^{11}}$ as
$\Rightarrow {x^{11}} = {x^{18 - 7}} \\ \Rightarrow {x^{11}} = {x^{18}} \times {x^{ - 7}}\,{\text{ }}\left[ {\because {x^{a + b}} = {x^a} \times {x^b}} \right] \\ \Rightarrow {x^{11}} = {x^{18}} \times \dfrac{1}{{{x^7}}}{\text{ }}\left[ {\because {x^{ - a}} = \dfrac{1}{{{x^a}}}} \right] \\ \therefore {x^{11}} = \dfrac{{{x^{18}}}}{{{x^7}}} \\$
So, the value ${x^{11}}$ is given by
$\Rightarrow {x^{11}} = \dfrac{{{x^{18}}}}{{{x^7}}} \\ \therefore {x^{11}} = \dfrac{{{m^2}}}{k}\,{\text{ }}\left[ {\because {x^{18}} = {m^2},{x^7} = k} \right] \\$
Thus, the correct option is A. $\dfrac{{{m^2}}}{k}$
Note: Here, we have used the formulae ${x^{a + b}} = {x^a} \times {x^b}$ and ${x^{ - a}} = \dfrac{1}{{{x^a}}}$. In these kinds of questions, try to convert the required term in terms of given terms and then substitute them to solve easily.