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# The simplest rationalizing factor of $\sqrt{75}$​ isA) ${{\text{(75)}}^{1/3}}$B) $\text{5}\sqrt{3}$C) $\text{3}$D) $\sqrt{150}$

Last updated date: 07th Aug 2024
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Hint: Here we apply the concept of prime factorization.
Prime number means a number with exactly two factors i.e., $1$ and itself
Prime factorization means expressing the number as a product of two or more prime numbers.
Rationalizing Factor: The multiplication factor which converts irrational number to rational number.
We need to find the simplest rationalizing factor of $\sqrt{75}$​.

To find the simplest rationalization factor of $\sqrt{75}$ we need to prime factorize $75$.
$75=5\times 5\times 3$
\begin{align} & \Rightarrow 75={{5}^{2}}\times 3 \\ & \sqrt{75}=\sqrt{{{5}^{2}}\times 3} \\ & \Rightarrow \sqrt{75}=5\sqrt{3} \\ \end{align}
\begin{align} & \sqrt{ab}=\sqrt{a}\times \sqrt{b} \\ & \sqrt{{{a}^{2}}}=a \\ \end{align}
For example: number $\text{ n=}{{\text{a}}^{p}}\times {{b}^{q}}\times {{c}^{r}}$ , where $a,b\text{ }\,and\,\text{ }c$ are prime numbers.