Question

How do you find the square root of $12025$?

Answer 1

Hint: Here we will try to simplify the given square root and then find the approximate value of the root by using the Babylonian method. After doing some simplification we get the required answer.

Complete step-by-step solution:
The given question, we can write it as,
$ \Rightarrow \sqrt {12025} $
Now the number $12025$ can be written as multiplication of the following terms:
$12025 = 5 \times 5 \times 481$
Now since the number $5$ is present twice in the multiplication, we can remove it out of the root part and write the equation as:
$ \Rightarrow 5\sqrt {481} \to (1)$
Now we will find the rational approximations for $\sqrt {481} $ using the Babylonian method as follows:
We know that given a rational approximation in the form $\dfrac{p}{q}$ to $\sqrt n $, we can find a better improved approximation by calculating the value of $\dfrac{{{p^2} + n{q^2}}}{{2pq}}$
Now, in our given problem the number $481$ is close to the number $484$ which has a square root value of $22$ therefore for the first approximation we will use $\dfrac{p}{q}$ as $\dfrac{{22}}{1}$
Here \[n = 481\]
Now the next approximation can be calculated as:
$ \Rightarrow \dfrac{{{{22}^2} + 481 \times {1^2}}}{{2 \times 22 \times 1}}$
On simplifying we get:
$ \Rightarrow \dfrac{{484 + 481}}{{44}}$
This can be written as:
$ \Rightarrow \dfrac{{965}}{{44}}$
Now to get a more accurate answer, we will use $\dfrac{p}{q}$ as $\dfrac{{965}}{{44}}$ in the next approximation.
Now the next approximation can be calculated as:
$ \Rightarrow \dfrac{{{{965}^2} + 481 \times {{44}^2}}}{{2 \times 965 \times 44}}$
On simplifying we get:
$ \Rightarrow \dfrac{{931225 + 931216}}{{84920}}$
This can be written as:
$ \Rightarrow \dfrac{{1862441}}{{84920}}$
On simplifying we get:
$ \Rightarrow \dfrac{{1862441}}{{84920}} \approx 21.9317$
Now on substituting this value in equation $(1)$we get:
$ \Rightarrow 5 \times 21.9317$
On simplifying we get:
$ \Rightarrow 109.6585$

Therefore, $\sqrt {12025} = 109.6585$

Note: It is to be remembered that the solution from the Babylonian method is an approximate value of the square root and not the actual value.
The actual answer might vary from the approximate value when calculated using a calculator.
The square root of a number can also be represented as a power in the form ${x^{\dfrac{1}{2}}}$.