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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

Last updated date: 16th May 2024
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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.

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, $\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}$
$\left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right) = {\left( {\sqrt 3 + \sqrt {10} } \right)^2} - {\left( {\sqrt 5 } \right)^2} \\ \left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right) = 3 + 10 + 2\sqrt 3 \sqrt {10} - 5 \\ \left( {\sqrt 3 + \sqrt {10} - \sqrt 5 } \right)\left( {\sqrt 3 + \sqrt {10} + \sqrt 5 } \right) = 8 + 2\sqrt {30} \\$
$\left( {8 + 2\sqrt {30} } \right)\left( {8 - 2\sqrt {30} } \right) = {8^2} - {\left( {2\sqrt {30} } \right)^2} \\ \left( {8 + 2\sqrt {30} } \right)\left( {8 - 2\sqrt {30} } \right) = 64 - 4 \times 30 \\ \left( {8 + 2\sqrt {30} } \right)\left( {8 - 2\sqrt {30} } \right) = 64 - 120 = - 56 \\$
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.