Question

How do you find the square root of 1369?

Answer 1

Hint: First write the given number in exponential form and then expand the given number into factors. If the given number is difficult to be broken down into smaller prime factors, use the long division method to find the value of the square root.

Complete step-by-step solution:
The given question is to find $\sqrt{1369}$
We know that The Law of Radicals is derived from the Laws of exponents. The expression $\sqrt[n]{a}$, n is called index,$\sqrt{{\text{ }}}$is called radical, and $a$ is called the radicand.$\sqrt[n]{a}={{a}^{\dfrac{1}{n}}}$The left side of the equation is known as radical form and the right side is exponential form.
A Radical represents a fractional exponent in which the numerator and the denominator of the radical contains the power of the base and the index of the radical, respectively.
Now rewrite our question.
$\Rightarrow x={{1369}^{\dfrac{1}{2}}}$
Now square on both sides of the expression.
$\Rightarrow {{x}^{2}}=1369$
We have to find a number which on squared by itself results in this number, 1369.
We can try to expand this number but as we can see this number is not divisible by any of the smaller primes like 2,3,5,7 etc.
So, we use the long division method to solve this.
Firstly, we group the number in two sets, 13 and 69 from right to left.
And now Let us start with 13.
The largest square which is less than or equal to 13 is 9 hence we divide with 3.
$3\left| \!{\underline {\,
  \begin{align}
  & 1369 \\
 & -9 \\
\end{align} \,}} \right. $
Now subtract 9 from 13 and write it as whole and write the second set which is 69 as it is.
$3\left| \!{\underline {\,
  \begin{align}
  & 1369 \\
 & -9 \\
 & 469 \\
\end{align} \,}} \right. $
Now double the divisor we have till now, which is 3 and find which number to be placed in these question marks.
$6?\times ?=469$
Now the smallest number which satisfies this is 7.
We write this number in the divisor along with 3.
$67\left| \!{\underline {\,
  \begin{align}
  & 469 \\
 & -469 \\
 & 0 \\
\end{align} \,}} \right. $
Now this results in zero.
Hence our obtained divisor is 37.
Therefore, the square root of 1369 is 37.

Note: Our solution is 37. It is a prime number. It would be difficult to expand to write it in exponential form or guess while finding the square root of the number 1369. Hence it would be easier to use this long division method to find out.