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Find the rationalizing factor of: \[\left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\]
A. \[\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right)\left( {8 - 2\sqrt {30} } \right)\]
B. \[\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right)\]
C. \[\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right)\left( {8 + 2\sqrt {30} } \right)\]
D. None of the above

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Last updated date: 16th May 2024
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Answer
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Hint: The factor of multiplication by which rationalization is done is called the rationalizing factor. So, rationalize the given expression by using the formula \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\] twice to obtain the required answer.

Complete step-by-step answer:
Rationalising factor means the term by which we convert irrational numbers to rational numbers.
So, it means we have to choose a term by which we make \[\left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\] is a rational number.
We know that, \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\]
Since,
 \[
  \left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right) = {\left( {\sqrt 3 + \sqrt {10} } \right)^2} - {\left( {\sqrt 5 } \right)^2} \\
  \left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right) = 3 + 10 + 2\sqrt 3 \sqrt {10} - 5 \\
  \left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right) = 8 + 2\sqrt {30} \\
\]
And
\[
  \left( {8 + 2\sqrt {30} } \right)\left( {8 - 2\sqrt {30} } \right) = {8^2} - {\left( {2\sqrt {30} } \right)^2} \\
  \left( {8 + 2\sqrt {30} } \right)\left( {8 - 2\sqrt {30} } \right) = 64 - 4 \times 30 \\
  \left( {8 + 2\sqrt {30} } \right)\left( {8 - 2\sqrt {30} } \right) = 64 - 120 = - 56 \\
\]
Therefore, \[\left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right)\left( {8 - 2\sqrt {30} } \right) = - 56\] which is a rational number.
Hence, \[\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right)\left( {8 - 2\sqrt {30} } \right)\] is a rational number \[\left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\].
Thus, the correct option is A. \[\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right)\left( {8 - 2\sqrt {30} } \right)\]

Note: In this question, the given expression \[\left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\] is a Surd. If the product of two or more surds is a rational number then they are rationalizing factors to each other. Sometimes we divide to get the rationalizing factor.