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

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Last updated date: 11th Jun 2024
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Answer
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Hint: Here, we will rewrite the given equation by using the rule \[{a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}\] and
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]. Then simplify the obtained equation to find the required value.

Complete step-by-step answer:
We are given that \[\sqrt {34 + 24\sqrt 2 } \times \left( {4 - 3\sqrt 2 } \right)\].
Rewriting the given equation, we get
\[
   \Rightarrow \sqrt {16 + 18 + 2 \cdot 4 \cdot 3\sqrt 2 } \times \left( {4 - 3\sqrt 2 } \right) \\
   \Rightarrow \sqrt {{4^2} + {{\left( {3\sqrt 2 } \right)}^2} + 2 \cdot 4 \cdot 3\sqrt 2 } \times \left( {4 - 3\sqrt 2 } \right) \\
 \]
Using the rule,\[{a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}\] in the above equation, we get
\[
   \Rightarrow \sqrt {{{\left( {4 + 3\sqrt 2 } \right)}^2}} \times \left( {4 - 3\sqrt 2 } \right) \\
   \Rightarrow \left( {4 + 3\sqrt 2 } \right) \times \left( {4 - 3\sqrt 2 } \right) \\
 \]
Using the rule,\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] in the above equation, we get
\[
   \Rightarrow {4^2} - {\left( {3\sqrt 2 } \right)^2} \\
   \Rightarrow 16 - 18 \\
   \Rightarrow - 2 \\
 \]
Thus, the required value is \[ - 2\].
Hence, option B is the correct answer.

Note: In this question, students should know that a square root of a number is a value that, when multiplied by itself, gives the number. This is a really simple problem, basic knowledge about the trigonometric is enough. We need to know that when \[\sqrt 2 \] is multiplied with itself is equal to 2.