Question

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. \[\sqrt {\left( {75 - {{10}^2}} \right)} = 5\]

Answer 1

Hint: Here we have to choose the option in which the given mathematical expression is not correct. So, we will first convert the square root in the power form and then we will use different properties of an exponential function to simplify the terms, and then after simplifying the terms, we will get the required answer.

Complete step by step solution:
Here we have to choose the option in which the given mathematical expression is not correct.
We will first consider the mathematical expression in option A i.e. \[\sqrt {50} = 5 \times \sqrt 2 \]
We will consider the left-hand expression.
LHS \[ = \sqrt {50} \]
Now, we will convert the square root into the power form.
\[ \Rightarrow \] LHS \[ = {\left( {50} \right)^{\dfrac{1}{2}}}\]
Now, we will write the number 50 as the product of its factors.
\[ \Rightarrow \] LHS \[ = {\left( {2 \times 5 \times 5} \right)^{\dfrac{1}{2}}}\]
Now, we will use the property of exponential function that when the exponentials with the same base are multiplied then their powers get added.
\[ \Rightarrow \] LHS \[ = {\left( {2 \times {5^2}} \right)^{\dfrac{1}{2}}}\]
On further simplification, we get
\[ \Rightarrow \] LHS \[ = {2^{\dfrac{1}{2}}} \times {5^{2 \times \dfrac{1}{2}}}\]
On further simplifying the powers, we get
\[ \Rightarrow \] LHS \[ = {2^{\dfrac{1}{2}}} \times 5\]
Now, we will again write the power in root form.
\[ \Rightarrow \] LHS \[ = 5 \times \sqrt 2 = \] RHS
So the expression in option A is correct.
Now, we will first consider the mathematical expression in option A i.e. \[\sqrt {60} = 4 \times \sqrt {15} \]
We will consider the left-hand expression.
LHS \[ = \sqrt {60} \]
Now, we will convert the square root into the power form.
\[ \Rightarrow \] LHS \[ = {\left( {60} \right)^{\dfrac{1}{2}}}\]
Now, we will write the number 50 as the product of its factors.
\[ \Rightarrow \] LHS \[ = {\left( {2 \times 2 \times 3 \times 5} \right)^{\dfrac{1}{2}}}\]
Now, we will use the property of exponential function that when the exponentials with the same base are multiplied then their powers get added.
\[ \Rightarrow \] LHS \[ = {\left( {{2^2} \times 3 \times 5} \right)^{\dfrac{1}{2}}}\]
On further simplification, we get
\[ \Rightarrow \] LHS \[ = {2^{2 \times \dfrac{1}{2}}} \times {\left( {3 \times 5} \right)^{\dfrac{1}{2}}}\]
On further simplifying the powers, we get
\[ \Rightarrow \] LHS \[ = 2 \times {\left( {15} \right)^{\dfrac{1}{2}}}\]
Now, we will again write the power in root form.
\[ \Rightarrow \] LHS \[ = 2 \times \sqrt {15} \ne \] RHS
So the expression in option B is incorrect.
Now, we will first consider the mathematical expression in option C i.e. \[\sqrt {243} = 9 \times \sqrt 3 \]
We will consider the left-hand expression.
LHS \[ = \sqrt {243} \]
Now, we will convert the square root into the power form.
\[ \Rightarrow \] LHS \[ = {\left( {243} \right)^{\dfrac{1}{2}}}\]
Now, we will write the number 50 as the product of its factors.
\[ \Rightarrow \] LHS \[ = {\left( {3 \times 3 \times 3 \times 3 \times 3} \right)^{\dfrac{1}{2}}}\]
Now, we will use the property of exponential function that when the exponentials with the same base are multiplied then their powers get added.
\[ \Rightarrow \] LHS \[ = {\left( {{3^4} \times 3} \right)^{\dfrac{1}{2}}}\]
On further simplification, we get
\[ \Rightarrow \] LHS \[ = {3^{4 \times \dfrac{1}{2}}} \times {\left( 3 \right)^{\dfrac{1}{2}}}\]
On further simplifying the powers, we get
\[ \Rightarrow \] LHS \[ = {3^2} \times {3^{\dfrac{1}{2}}}\]
Now, we will again write the power in root form.
\[ \Rightarrow \] LHS \[ = 9 \times \sqrt 3 = \] RHS
So the expression in option C is correct.
Now, we will first consider the mathematical expression in option D i.e. \[\sqrt {\left( {75 - {{10}^2}} \right)} = 5\]
We will consider the left-hand expression.
LHS \[ = \sqrt {\left( {75 - {{10}^2}} \right)} \]
We will first simplify the terms under the square root.
\[ \Rightarrow \] LHS \[ = \sqrt {\left( {75 - 100} \right)} \]
\[ \Rightarrow \] LHS \[ = \sqrt { - 25} \ne \] RHS
So the expression in option D is incorrect.

Hence, option B and option D are incorrect.

Note: Here we have used the properties of the exponential function. Here exponential functions are defined as the functions which are inverse of the logarithmic functions i.e. when taking the inverse of the exponential functions, we will get the logarithmic function.