Question

Which of the following is not true?
This question has multiple correct options.
(a) $\sqrt{50}=5\times \sqrt{2}$
(b) $\sqrt{60}=4\times \sqrt{15}$
(c) $\sqrt{243}=9\times \sqrt{3}$
(d) $\sqrt{75-{{10}^{2}}}=5$

Answer 1

Hint: For solving this question we will simply square the numbers on the left-hand side and right-hand side in each part. After that, we will see whether the number on the left-hand side is equal to the right-hand side and select the correct options as per the given question.

Complete step-by-step solution -
Given:
We have to find which of the following is not true?
(a) $\sqrt{50}=5\times \sqrt{2}$
(b) $\sqrt{60}=4\times \sqrt{15}$
(c) $\sqrt{243}=9\times \sqrt{3}$
(d) $\sqrt{75-{{10}^{2}}}=5$
Now, before we proceed we should know that, if $a>0$ then, $\sqrt{a}$ will be real otherwise $\sqrt{a}$ will be imaginary.
Now, we will check each option.
(a) $\sqrt{50}=5\times \sqrt{2}$
Now, we will square each number on the left-hand side and right-hand side. Then,
$\begin{align}
  & {{\left( \sqrt{50} \right)}^{2}}=50 \\
 & {{\left( 5\times \sqrt{2} \right)}^{2}}={{5}^{2}}\times 2=25\times 2=50 \\
 & \Rightarrow LHS=RHS \\
\end{align}$
Now, from the above result, we conclude that, $\sqrt{50}=5\times \sqrt{2}$ is true.

(b) $\sqrt{60}=4\times \sqrt{15}$
Now, we will square each number on the left-hand side and right-hand side. Then,
$\begin{align}
  & {{\left( \sqrt{60} \right)}^{2}}=60 \\
 & {{\left( 4\times \sqrt{15} \right)}^{2}}={{4}^{2}}\times 15=16\times 15=240 \\
 & \Rightarrow LHS\ne RHS \\
\end{align}$
Now, from the above result, we conclude that $\sqrt{60}=4\times \sqrt{15}$ is not true.

(c) $\sqrt{243}=9\times \sqrt{3}$
Now, we will square each number on the left-hand side and right-hand side. Then,
$\begin{align}
  & {{\left( \sqrt{243} \right)}^{2}}=243 \\
 & {{\left( 9\times \sqrt{3} \right)}^{2}}={{9}^{2}}\times 3=81\times 3=243 \\
 & \Rightarrow LHS=RHS \\
\end{align}$
Now, from the above result, we conclude that $\sqrt{243}=9\times \sqrt{3}$ is true.

(d) $\sqrt{75-{{10}^{2}}}=5$
Now, before we square we should check whether $75-{{10}^{2}}$ is positive or not. Then,
$\begin{align}
  & 75-{{10}^{2}}=75-100 \\
 & \Rightarrow 75-{{10}^{2}}=-25 \\
 & \Rightarrow 75-{{10}^{2}}<0 \\
\end{align}$
Now, from the above result, we conclude that $75-{{10}^{2}}$ will be less than zero. Then, $\sqrt{75-{{10}^{2}}}$ will be an imaginary number.
Now, we have proved that in the left-hand side we have an imaginary number. Moreover, on the right-hand side, we have number 5 which is a real number. Then,
$LHS\ne RHS$
Now, from the above result, we conclude that, $\sqrt{75-{{10}^{2}}}=5$ is not true.
Now, as we have to select options which are not true.
Hence, options (b) and (d) will be the correct options.

Note: Here, the student should first read the question clearly then, simply check for each option carefully. Moreover, though the question is very easy, we should be careful while selecting the options and we should select options in which the given equation is not true.