Question

What numbers does the square root of \[45\] lie in?

Answer 1

Hint: The inverse (opposite) operation of addition is subtraction and the inverse operation of multiplication is division. Similarly, finding the square root is the inverse operation of squaring.
 We have,
 \[{1^2} = 1\], therefore square root of $1$ is $1$
\[{2^2} = 4\], therefore square root of $4$is $2$
\[{3^2} = 9\], therefore square root of $9$ is$3$
To find: Square root of \[45\]

Complete step-by-step solution:
Step 1: Place a bar over every pair of digits starting from the digit at one’s place. Thus we have, \[\overline {45} \]
Step 2: Find the largest number whose square is less than or equal to the number under the bar, here we have \[({6^2} < 45 < {7^2})\]. Take this number as the divisor and the quotient with the number under the bar as the dividend (here\[45\]). Divide and get the remainder ($9$ in this case).
Step 3: Now take a pair of zeroes to the right of the remainder, so the new dividend is \[900\] and with this we have to add decimal to the right of the quotient.
Step 4: Double the quotient and enter it with a blank on its right.
Step 5: Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend. In this case, \[127 \times 7 = 889\]. Get the remainder
Step 6: Still we have remainder equal to \[11\] so we follow the same step for more exact value.
Hence, the square root of \[45\] is approximately equal to \[6 \cdot 7\].
Therefore, the square root of \[45\] lies in between
 \[6 < \sqrt {45} < 7\]

Note:
> If a natural number m can be expressed as.\[{n^2}\]., where n is also a natural number, and then $m$ is a square number.
> All square numbers end with \[0,1,4,5,6or9\] at the unit's place.
> Square numbers can only have an even number of zeros at the end.
> Square root is the inverse operation of square.