Question

Which of the following is correct?
(a)$\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
(b)$\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
(c)$\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
(d)$\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+6 \right)$

Answer 1

Hint: We can find the answer by expanding on both left hand side and right hand side. The equation whose LHS=RHS will be the right option. So take each option and simplify the LHS by multiplying the terms and expressing in simplest form.

Complete step by step solution:
First, taking the equation from option (a), we get,$\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
LHS becomes,
$\begin{align}
  & LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}-24 \\
 & \Rightarrow LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right) \right\}-24 \\
 & \Rightarrow LHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24......\left( i \right) \\
\end{align}$
RHS becomes,
$\begin{align}
  & RHS=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-6\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-\left( 4\times 6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24......\left( ii \right) \\
\end{align}$
Now, comparing equations (i) and (ii), we can clearly see that $LHS=RHS$ , Therefore, the option (a) is the correct option.
Next, let us check the remaining options to make sure they are not correct.
 Taking the equation from option (b), we get, $\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
LHS becomes,
$\begin{align}
  & LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}+24 \\
 & \Rightarrow LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right) \right\}+24 \\
 & \Rightarrow LHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24......\left( iii \right) \\
\end{align}$
RHS becomes,
$\begin{align}
  & RHS=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-6\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-\left( 4\times 6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24......\left( iv \right) \\
\end{align}$
Now, comparing equations (iii) and (iv), we can clearly see that $LHS\ne RHS$ , Therefore, the option (b) is an incorrect option. Now, moving on to the next option,
Taking the equation from option (c), we get, $\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
LHS becomes,
$\begin{align}
  & LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}-24 \\
 & \Rightarrow LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right) \right\}-24 \\
 & \Rightarrow LHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24......\left( v \right) \\
\end{align}$
RHS becomes,
$\begin{align}
  & RHS=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-6\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+\left( 4\times 6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-10\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24......\left( vi \right) \\
\end{align}$
Now, comparing equations (v) and (vi), we can clearly see that $LHS\ne RHS$ , Therefore, the option (c) is an incorrect option. Now, moving on to the final option,
Taking the equation from option (d), we get, $\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+6 \right)$
LHS becomes,
$\begin{align}
  & LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}+24 \\
 & \Rightarrow LHS=\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right) \right\}+24 \\
 & \Rightarrow LHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24......\left( vii \right) \\
\end{align}$
RHS becomes,
$\begin{align}
  & RHS=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+6\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-\left( 4\times 6 \right) \\
 & \Rightarrow RHS=\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24......\left( viii \right) \\
\end{align}$
Now, comparing equations (vii) and (viii), we can clearly see that $LHS\ne RHS$ , Therefore, the option (d) is an incorrect option.

So, the correct answer is “Option A”.

Note: We can solve the question in a simpler way by this method. Substitute some constant A as $\sqrt[3]{7}+\sqrt[7]{3}$. That is $A=\sqrt[3]{7}+\sqrt[7]{3}$ . Now, using this value of A, we can check the options.
First take option (a), use A wherever necessary.
$\begin{align}
  & \left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & \Rightarrow \left\{ \left( A \right)\left( A-2 \right) \right\}-24=\left( A+4 \right)\left( A-6 \right) \\
 & \Rightarrow {{A}^{2}}-2A-24={{A}^{2}}-6A+4A-24 \\
 & \Rightarrow {{A}^{2}}-2A-24={{A}^{2}}-2A-24 \\
 & \Rightarrow LHS=RHS \\
\end{align}$
Therefore, the option (a) is the correct option. Next, let us check the remaining options to make sure they are not correct.
Take option (b), use A wherever necessary.
$\begin{align}
  & \left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & \Rightarrow \left\{ \left( A \right)\left( A+2 \right) \right\}+24=\left( A+4 \right)\left( A-6 \right) \\
 & \Rightarrow {{A}^{2}}+2A+24={{A}^{2}}-6A+4A-24 \\
 & \Rightarrow {{A}^{2}}+2A+24={{A}^{2}}-2A-24 \\
 & \Rightarrow LHS\ne RHS \\
\end{align}$
Therefore, the option (b) is not a correct option.
Take option (c), use A wherever necessary.
$\begin{align}
  & \left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & \Rightarrow \left\{ \left( A \right)\left( A+2 \right) \right\}-24=\left( A-4 \right)\left( A-6 \right) \\
 & \Rightarrow {{A}^{2}}+2A-24={{A}^{2}}-6A-4A+24 \\
 & \Rightarrow {{A}^{2}}+2A-24={{A}^{2}}-10A+24 \\
 & \Rightarrow LHS\ne RHS \\
\end{align}$
Therefore, the option (c) is not a correct option.
Take option (d), use A wherever necessary.
$\begin{align}
  & \left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+6 \right) \\
 & \Rightarrow \left\{ \left( A \right)\left( A-2 \right) \right\}+24=\left( A-4 \right)\left( A+6 \right) \\
 & \Rightarrow {{A}^{2}}-2A+24={{A}^{2}}+6A-4A-24 \\
 & \Rightarrow {{A}^{2}}-2A+24={{A}^{2}}+2A-24 \\
 & \Rightarrow LHS\ne RHS \\
\end{align}$
Therefore, the option (d) is not a correct option.