Question

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

Answer 1

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 $\sqrt{75}$​.

Complete step by step answer:
To find the simplest rationalization factor of $\sqrt{75}$ we need to prime factorize $75$.
$75=5\times 5\times 3$
$\begin{align}
  & \Rightarrow 75={{5}^{2}}\times 3 \\
 & \sqrt{75}=\sqrt{{{5}^{2}}\times 3} \\
 & \Rightarrow \sqrt{75}=5\sqrt{3} \\
\end{align}$

Hence, the correct answer is option B.

Additional Information:
$\begin{align}
  & \sqrt{ab}=\sqrt{a}\times \sqrt{b} \\
 & \sqrt{{{a}^{2}}}=a \\
\end{align}$
A number n can always be expressed as a product of primes. The least prime is 2.
For example: number $\text{ n=}{{\text{a}}^{p}}\times {{b}^{q}}\times {{c}^{r}}$ , where $a,b\text{ }\,and\,\text{ }c$ are prime numbers.

Note: In such type of questions which involves concept of rationalization most of the times prime factorization becomes the key factor to find the answer required. We will need to have an idea of doing prime factorization.