Question

If the value of $\sqrt {35 - 12\sqrt 6 - 4\sqrt {15} + 6\sqrt {10} } = m\sqrt 2 - k\sqrt 3 + \sqrt 5 $, then find the value of $(m+k)$.

Answer 1

Hint: The general algebraic properties will be used in this question. We will square both the sides and compare the different terms to find the value of m and k. The property to be used is-
${\left( {{\text{a}} + {\text{b}} + {\text{c}}} \right)^2} = {{\text{a}}^2} + {{\text{b}}^2} + {{\text{c}}^2} + 2ab + 2ab + 2ca$...(1)

Complete step-by-step answer:
We have been given that-
$\sqrt {35 - 12\sqrt 6 - 4\sqrt {15} + 6\sqrt {10} } = m\sqrt 2 - k\sqrt 3 + \sqrt 5 $
On squaring both the sides of the equation and using property (1) we get,
$\begin{align}
  &35 - 12\sqrt 6 - 4\sqrt {15} + 6\sqrt {10} = {\left( {{\text{m}}\sqrt 2 - {\text{k}}\sqrt 3 + \sqrt 5 } \right)^2} \\
  &35 - 12\sqrt 6 - 4\sqrt {15} + 6\sqrt {10} = 2{{\text{m}}^2} + 3{{\text{k}}^2} + 5 - 2mk\sqrt 6 - 2{\text{k}}\sqrt {15} + 2{\text{m}}\sqrt {10} \\
  &35 - 12\sqrt 6 - 4\sqrt {15} + 6\sqrt {10} = \left( {2{{\text{m}}^2} + 3{{\text{k}}^2} + 5} \right) - 2mk\sqrt 6 - 2{\text{k}}\sqrt {15} + 2{\text{m}}\sqrt {10} \\
\end{align} $

By comparing the terms in the two sides, we can equate the coefficients of the square root terms to form equations in m and k, and then solve them to find their respective values.

By comparing coefficients of $\sqrt {15}$ and $\sqrt {10}$,
$\begin{align}
   &- 4 = - 2{\text{k}}\;and\;6 = 2{\text{m}} \\
  &{\text{k}} = 2\;and\;{\text{m}} = 3 \\
\end{align} $

We can further check and verify these values by comparing the coefficients of $\sqrt 6 $,
$\begin{align}
   &- 12 = - 2mk \\
  &mk = 6 \\
  &{\text{k}} = 2\;and\;{\text{m}} = 3\;so, \\
  &3 \times 2 = 6 \\
 & 6 = 6 \\
\end{align} $

Hence, the answer is verified. We need to find the value of $k + m$, so
$m + k = 3 + 2 = 5$
This is the required answer.

Note: A good method to check your answers in such types of questions are by substituting the values obtained in the third equation that is formed. In fact, in this question, there are four equations possible between k and m, for each pair of terms. We can select any two pairs to solve, but it is always advisable to select those which are in the simplest form.