Question

The value of the expression\[\sqrt {34 + 24\sqrt 2 } \times (4 - 3\sqrt 2 )\] is?
A. - 4
B. - 2
C. 3
D. 4

Answer 1

Hint: In order to solve this problem we first keep \[\sqrt {{{(4 - 3\sqrt 2 )}^2}} \]in place of \[(4 - 3\sqrt 2 )\]so that the problem can be simplified easily, and then do further calculations.

Complete step-by-step answer:
To start with, we have,
\[\sqrt {34 + 24\sqrt 2 } \times (4 - 3\sqrt 2 )\]
using, \[\sqrt {{{(4 - 3\sqrt 2 )}^2}} \]in place of \[(4 - 3\sqrt 2 )\], we get,
\[ = \sqrt {34 + 24\sqrt 2 } \times \sqrt {{{(4 - 3\sqrt 2 )}^2}} \]
On simplifying further, we get,
\[ = \sqrt {(34 + 24\sqrt 2 )(16 + 18 - 24\sqrt 2 )} \]
\[ = \sqrt {(34 + 24\sqrt 2 )(34 - 24\sqrt 2 )} \]
Now we use the formula, \[{a^2} - {b^2} = (a + b)(a - b)\], so we get,
\[ = \sqrt {{{(34)}^2} - {{(24\sqrt 2 )}^2}} \]
\[ = \sqrt {1156 - 1152} \]
\[ = \sqrt 4 = \pm 2\]
As per our options, we can see that the correct option is (B)

Note: The problem can also be solved by first finding the value of \[\sqrt {34 + 24\sqrt 2 } \]. But that would be a tougher and longer process.