Simplify \[\sqrt {10\dfrac{{151}}{{225}}} \].

Hint: In the given question, we have to find the square root of a mixed number \[\sqrt {10\dfrac{{151}}{{225}}} \]. In order to solve it, first we will have to convert the number into an improper fraction and then find out it’s square root easily.
A mixed number is a combination i.e. a mixture of the whole number and a fraction expressed together as a single number. It is partly a whole number and partly a fraction.

Complete step by step solution:
The fraction can be proper or improper. Proper fraction is the one where the numerator is less than the denominator whereas improper fraction is the one where numerator is more than the denominator.
Examples of mixed number are - \[12\dfrac{3}{{20}}\],\[7\dfrac{5}{{16}}\], \[8\dfrac{2}{7}\] etc.
We can convert a mixed number into improper fraction by following the below given steps:
Step One: Multiply the whole number by the denominator
Step Two: Add the answer from Step One to the numerator
Step Three: Write the answer of Step Two over the denominator.
We can solve the given sum as follows:
First converting \[10\dfrac{{151}}{{225}}\] into improper fraction-
Step One: Multiplying \[10\]with the denominator of improper fraction \[225\] i.e. \[10 \times 225\]
\[ = 2250\]
Step Two: Adding the above answer to the numerator of the improper fraction i.e. \[2250 + 151\]
\[ = 2401\]
Step Three: Forming the improper fraction with above answer as numerator i.e. \[\dfrac{{2401}}{{225}}\]
Hence, we have converted the mixed number into improper fraction, \[10\dfrac{{151}}{{225}} = \dfrac{{2401}}{{225}}\].
Now we can find the square root by prime factorization method since they are perfect squares:
\[\sqrt {\dfrac{{2401}}{{225}}} \]
=\[\sqrt {\dfrac{{49 \times 49}}{{15 \times 15}}} \]\[ = \sqrt {\dfrac{{{{(49)}^2}}}{{{{(15)}^2}}}} \]
\[ = \dfrac{{49}}{{15}}\]
Therefore, \[\sqrt {10\dfrac{{151}}{{225}}} = \dfrac{{49}}{{15}}\].

Note:
> To find the square root of any number, we need to check whether the given number is a perfect square or not.
> If it is a perfect square then we can solve it by prime factorization method.
> If it is not a perfect square, then we will need to use the division method.