Question

Which one 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: The question is pretty simple we just have to multiply the left hand and right hand side of the options given in the question but the multiplication done in such a manner that we are going to multiply $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ with $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ without opening the brackets i.e. multiplication of these two root over terms is ${{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}$. Write this square expression as it doesn't open this square terms on both sides of the equation.

Complete step-by-step solution -
First of all we are checking the option (a) by multiplying on both the sides of the equation.
(a) $\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
Solving the left hand side of the above equation we get,
$\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}-24$
Multiplying $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ first with $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ then multiply $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ with -2 we get,
$\begin{align}
  & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
\end{align}$
Solving the right hand side of the equation given in option (a) we get,
$\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
Multiplying $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ first with $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ then multiply $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ with -6 then multiply $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ with 4 and then multiply 4 with -6 we get,
$\begin{align}
  & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-6\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
\end{align}$
As you can see from the above that L.H.S is equal to R.H.S so this option is correct.

(b) $\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
Solving the left hand side of the above equation we get,
\[\begin{align}
  & \left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}+24 \\
 & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24 \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24 \\
\end{align}\]
Solving the right hand side of the equation given in option (b) we get,
$\begin{align}
  & \left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-6\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
\end{align}$
As you can see from the above that L.H.S is not equal to R.H.S so this option is not correct.

(c) $\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}-24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right)$
Solving the left hand side of the above equation we get,
$\begin{align}
  & \left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+2 \right) \right\}-24 \\
 & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
\end{align}$
Solving the right hand side of the equation given in option (c) we get,
$\begin{align}
  & \left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-6\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24 \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}-10\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24 \\
\end{align}$
As you can see from the above that L.H.S is not equal to R.H.S so this option is not correct.

(d) $\left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}+24=\left( \sqrt[3]{7}+\sqrt[7]{3}-4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}+6 \right)$
Solving the left hand side of the above equation we get,
$\begin{align}
  & \left\{ \left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-2 \right) \right\}+24 \\
 & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24 \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24 \\
\end{align}$
Solving the right hand side of the equation given in option (d) we get,
$\begin{align}
  & \left( \sqrt[3]{7}+\sqrt[7]{3}+4 \right)\left( \sqrt[3]{7}+\sqrt[7]{3}-6 \right) \\
 & =\left( \sqrt[3]{7}+\sqrt[7]{3} \right)\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+4\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-6\left( \sqrt[3]{7}+\sqrt[7]{3} \right) \\
 & ={{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24 \\
\end{align}$
As you can see from the above that L.H.S is not equal to R.H.S so this option is not correct.
From the checking of the options, we have found that option (a) is correct.

Note: In the above solution you have observed that we have not opened the square of $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ because if you open it then unnecessarily you will make the calculations complex and then you will be confused whether the left hand and right hand side of the option is correct or not.
In the below, we are demonstrating for option (b)
The result of the left hand side of the option (b) is:
\[{{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}+2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)+24\]
Opening the square of $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ in the above expression we get,
${{7}^{\dfrac{2}{3}}}+{{3}^{\dfrac{2}{7}}}+2\sqrt[3]{7}\left( \sqrt[7]{3} \right)+2\sqrt[3]{7}+2\left( \sqrt[7]{3} \right)+24$ …………. Eq. (1)
The result of the right hand side of the option (b) is:
${{\left( \sqrt[3]{7}+\sqrt[7]{3} \right)}^{2}}-2\left( \sqrt[3]{7}+\sqrt[7]{3} \right)-24$
Opening the square of $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ in the above expression we get,
${{7}^{\dfrac{2}{3}}}+{{3}^{\dfrac{2}{7}}}+2\sqrt[3]{7}\left( \sqrt[7]{3} \right)-2\sqrt[3]{7}-2\left( \sqrt[7]{3} \right)-24$………….. Eq. (2)
On comparing eq. (1) and eq. (2) we can see that the first 3 terms are the same after that the terms differ by negative sign.
But you can see as compared to the calculations that we have done in the solution part when we have opened the square of $\left( \sqrt[3]{7}+\sqrt[7]{3} \right)$ then the calculations become more rigorous and comparison of the left hand and right hand side is taking time.