Question

The mean proportional between \[\sqrt {11} - \sqrt 5 \] and \[13\sqrt {11} + 19\sqrt 5 \]

Answer 1

Hint: To get the mean proportional of these two given numbers, use the concept that mean proportional of two numbers is the square root of the product of the two numbers.

Complete step-by-step answer:
Here the two numbers is given as \[\sqrt {11} - \sqrt 5 \] and \[13\sqrt {11} + 19\sqrt 5 \]
And we know that mean proportional of two numbers is the square root of the product of the two numbers
So, on applying the concept
We get,
 \[ = \sqrt {\left( {\sqrt {11} - \sqrt {5} } \right)\left( {13\sqrt {11} + 19\sqrt 5 } \right)} \]
now multiplying the two numbers
 \[ = \sqrt {143 - 95 - 13\sqrt {55} + 19\sqrt {55} } \]
or,
 \[ = \sqrt {48 + 6\sqrt {55} } \]
Hence the mean proportional of two given numbers is \[ = \sqrt {48 + 6\sqrt {55} } \]
So, the correct answer is “ \[ = \sqrt {48 + 6\sqrt {55} } \] ”.

Note: Here according to the solution we can say that the mean proportional of two numbers only defined if both the numbers are negative or both the numbers are positive.
For one positive integer and one negative integer the product will be negative inside the root which is complex and not defined in real numbers.