Question

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. \[k + 4\]

Answer 1

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.

Complete step-by-step answer:
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.