Question

Find the square root of \[{\mathbf{1057}}\].

Answer 1

Hint: Square: The positive number, when multiplied by itself, represents the square of the number.
Square root: The square root of any number is equal to a number, which when squared gives the original number. Square root is the inverse operation of square.
Let us say m is a positive integer, such that
\[\sqrt {(m.m)} = \sqrt {({m^2})} = m\]
The square root symbol is usually denoted as \['\sqrt {} '\]
It is a symbol called a radical symbol.
The number under the radical symbol is called the radicand.
To find square roots we can use two methods. Prime factorisation and long division method.

Complete step-by-step answer:
Square root of \[1057\]
We will group the digit into two from the right. In this case, it would be group \[57\] first and then \[10\].
$\begin{array}{*{20}{c}}
  {}& & {32}&{.51}&{} \\
\hline
  3& & {10}&{57}&{} \\
  { + 3}& & { - 9}& \Downarrow &{} \\
\hline
  {62}& & 1&5&7 \\
  { + 2}& & 1&2&4 \\
\hline
  {645}& & 3&3&{00} \\
  { + 5}& & 3&2&{25} \\
\hline
  {6501}& & {}&{75}&{00} \\
  {}& & - &{65}&{01} \\
\hline
  {}& & {}&9&{99}
\end{array}$

Hence the quotient is \[32.51\] and is the root of \[1057\]

Note: The square root of a negative number represents a complex number.
\[\sqrt { - n} = \sqrt[i]{n}\,\,\,\,\,i < imaginary\,number\]
The two square root values can be multiplying
For Example, \[\sqrt 3 \]can be multiplied by \[\sqrt 2 \], then the result should be \[\sqrt {6.} \]
The square root of any negative number is not defined. Because the perfect square cannot be negative.
We can write \[\sqrt a \]\[ = {\left( a \right)^{\dfrac{1}{2}}}\], it means we replace square root symbol by \[\left( {\dfrac{1}{2}} \right)\] as the power of radicand.