Question

How do you find the square root of \[23\]?

Answer 1

Hint:
We have to find the square root of \[23\]. Since \[23\] is a prime number so we cannot find the square root by the prime factorization method. So firstly we find the nearest perfect square of \[23\]. Then we write \[23\] in the form of \[25\] then we take square root on both sides then we solve the Right-hand side and write it in the form of \[(1 + xn)\] and we solve it for \[{(1 + x)^n}\] And at least We will be able to find the square root.

Complete Step by step Solution:
we have to find the square root of \[23\]. As \[23\] is a prime number so find it by another method.
The nearest perfect square of \[23\] is \[25\].
So 23 can be written as
\[23 = 25 - 2\]
Taking square root in both side
\[ \Rightarrow \sqrt {23} = \sqrt {25 - 2} \]
\[ \Rightarrow {(23)^{\dfrac{1}{2}}} = {(25 - 2)^{\dfrac{1}{2}}}\]
Taking 25 common from right-hand-side
\[ \Rightarrow {(23)^{\dfrac{1}{2}}} = {(25)^{\dfrac{1}{2}}}{\left[ {1 - \dfrac{2}{{25}}} \right]^{\dfrac{1}{2}}}\]
Square root of \[25{\text{ }} = 5\]
\[ \Rightarrow {(23)^{\dfrac{1}{2}}} = 5{\left[ {1 - \dfrac{2}{{25}}} \right]^{\dfrac{1}{2}}}\]
The term in the right hand side is in the form of \[{(1 + x)^n}\].
And \[{(1 + x)^n}\]is approximately equal to \[(1 - \dfrac{1}{n}x)\]
So \[{(23)^{\dfrac{1}{2}}} \approx 5\left[ {1 - \dfrac{2}{{2 \times 25}}} \right]\]
\[
    \\
   \Rightarrow {(23)^{\dfrac{1}{2}}} \approx 5\left[ {1 - \dfrac{1}{{25}}} \right] \\
 \]
\[ \Rightarrow {(23)^{\dfrac{1}{2}}} \approx 5\left[ {1 - \dfrac{1}{{25}}} \right]\]
\[ \Rightarrow {(23)^{\dfrac{1}{2}}} \approx 5\left[ {1 - 0.04} \right] \approx 5\left[ {0.96} \right]\]
\[ \Rightarrow {(23)^{\dfrac{1}{2}}} \approx 5\left[ {1 - 0.04} \right] \approx 5\left[ {0.96} \right]\]
\[ \Rightarrow {(23)^{\dfrac{1}{2}}} \approx 4.8\]
This is the required result.

Note:
Prime numbers are those numbers that are divided by one and itself only. In the number theory, integer factorization is the decomposition of the composite number into that smaller integer. If their smaller integers are restricted to prime numbers only the process is called prime factorization. The square root of a number is The number which when multiplied by itself gives the number.
Example: When we multiply \[4\] by \[4\] we get \[16\] so \[4\] is the Square root of \[16\].