Question

What is the square root of $2.25$ ?

Answer 1

Hint: First, we will convert the given decimal numbers into the form of fractions, because taking the square root to the decimal numbers are a confusing method and hence, we will convert the given decimal to fraction using the base $10$ division method for one decimal point.
Hence the given decimal number can be written as in the fraction form as $2.25 = \dfrac{{225}}{{100}}$ and now it is easy to take the square root and then finally we act the division to get the required answer.

Complete step-by-step solution:
Since from the given information we have converted into fraction form as $2.25 = \dfrac{{225}}{{100}}$
Now we will find its prime factorization and then taking the square root we get the required answer.
If the given number is prime then we cannot find its prime factorization. Now to check if the given number is prime or composite.
Since the number can divide $5,25,...$ and hence which is the composite number.
Prime numbers are the numbers that are divisible by themselves and $1$ only or also known as the numbers whose factors are the given number itself.
But the composite numbers which are divisible by themselves, $1$ and also with some other numbers (at least one number other than $1$ and itself)
Every composite number can be represented in the form of prime factorization.
Thus, the number $225$ can be represented in the prime factorization as
$
  3\left| \!{\underline {\,
  {225} \,}} \right. \\
  3\left| \!{\underline {\,
  {75} \,}} \right. \\
  5\left| \!{\underline {\,
  {25} \,}} \right. \\
  5\left| \!{\underline {\,
  5 \,}} \right. \\
  1\left| \!{\underline {\,
  1 \,}} \right. \\
 $
Hence, we have rewritten as $225 = 3 \times 3 \times 5 \times 5$ (since $3,5$ are prime numbers)
Also, the number $100$ can be represented as
$
  5\left| \!{\underline {\,
  {100} \,}} \right. \\
  5\left| \!{\underline {\,
  {20} \,}} \right. \\
  2\left| \!{\underline {\,
  4 \,}} \right. \\
  2\left| \!{\underline {\,
  2 \,}} \right. \\
  1\left| \!{\underline {\,
  1 \,}} \right. \\
 $
which means $100 = 5 \times 5 \times 2 \times 2$ as prime factors
Hence, we have $\dfrac{{225}}{{100}} = \dfrac{{3 \times 3 \times 5 \times 5}}{{2 \times 2 \times 5 \times 5}}$
Now take the square root we get $\sqrt {\dfrac{{225}}{{100}}} = \sqrt {\dfrac{{3 \times 3 \times 5 \times 5}}{{2 \times 2 \times 5 \times 5}}} \Rightarrow \dfrac{{3 \times 5}}{{5 \times 2}}$ since $\sqrt {2 \times 2} = 2$
By the division and multiplication operation, we get $\sqrt {\dfrac{{225}}{{100}}} = \dfrac{{15}}{{10}} \Rightarrow 1.5$
Hence the square root of $2.25$ is $1.5$

Note: Since don’t write the number ${2^2} = 4$because $4$ is the composite number, and thus ${2^2},{3^2},{5^2}$ is the repeated prime number while in the process of prime factorization.
We can find whether the given number is prime or composite by the trial-and-error methods. Divide the number with the prime numbers less than the given number. if the number is exactly divisible by the prime number, it is the composite number, if not then it is the prime number.
The only even prime number is $2$ and all other prime numbers are odd.