Question

How do you find $2$ square root of $325$?

Answer 1

Hint: Given the numbers in the form of a fraction. First, we will find the factors of the terms at the denominator and the numerator. Then, cancel out the common factors at the numerator and denominator. Then, write the result in a simplified form. Also, we will try to factor the radicand further. If the factors of the term are not possible, then the result in simplified form.

Formula used:
The division property of the radicals is given as:
$\dfrac{{\sqrt x }}{{\sqrt y }} = \sqrt {\dfrac{x}{y}} $

Complete step-by-step answer:
First, we will write the given expression in mathematical form.
$ \Rightarrow 2\sqrt {325} $
Now, to find factors of the radical expression, divide $325$ by $5$.
$ \Rightarrow \dfrac{{325}}{5} = 65$
Again, we will divide the number $65$ by $5$.
$ \Rightarrow \dfrac{{65}}{5} = 13$
Now, $13$ is a prime number which is divided by itself only. So, the factors of the radical expression is written as:
$ \Rightarrow 325 = 1 \times 5 \times 5 \times 13$
Now, add the radical to the factors of the expression.
$ \Rightarrow 2\sqrt {1 \times 5 \times 5 \times 13} $
Now, we will find the perfect squares inside the radical expression.
$ \Rightarrow 2\sqrt {{1^2} \times {5^2} \times 13} $
Now, take out the perfect square factor of the expression.
$ \Rightarrow 2 \times 1 \times 5\sqrt {13} $
Multiplying the terms in the expression, we get:
$ \Rightarrow 10\sqrt {13} $
Here, $\sqrt {13} $ is not a perfect square of any number which means this number is in its simplified form.

Hence the value of $2$ square root of $325$ is $10\sqrt {13} $

Additional Information: The factors of a certain number are the numbers that can divide that number without leaving any remainder. Then, the number in the radical can be solved by finding the perfect squares of the number and extract the factors out of radicals. If the number is not a perfect square, then the radical expression cannot be reduced further.

Note:
In such types of questions students mainly get confused in applying the formula. As they don't know which formula they have to apply. So, when a certain number is given, then to simplify the expression, factorize the number using the prime factorization method. Then, find the perfect squares of the number and take out of the radical.