Question

What is the square root of \[41.7\], \[0.6781\] and \[0.8\]?

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 \[41.7\], \[0.6781\] and \[0.8\]

Complete step-by-step solution:
Square root of \[41.7\]
Step 1: Try to remove the decimal part and convert it into pair form, we get
\[41.7 = 4170 \times {10^{ - 2}}\]
So \[4170\] have two pairs of numbers.
Step 2: Place a bar over every pair of digits starting from the digit at one’s place. Thus we have, \[\overline {41} \] and \[\overline {70} \]
Step 3: Find the largest number whose square is less than or equal to the number under the extreme left bar \[({6^2} < 41 < {7^2})\]. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend (here\[41\]). Divide and get the remainder ($5$ in this case).
Step 4: Bring down the number under the next bar (i.e.,$70$ in this case) to the right of the remainder. So the new dividend is \[570\].
Step 5: Double the quotient and enter it with a blank on its right.
Step 6: 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, \[124 \times 4 = 496\]. As \[123 \times 3 = 369\] & \[125 \times 5 = 625\] and both are less than and greater than \[570\] respectively, we choose the digit as $4$. Get the remainder.
Step 7: Since the remainder is \[74\] and no digits are left in the given number, therefore,
            \[\sqrt {4170} \simeq 64\] (Slightly greater than \[64\])
Step 8: Determine the value of \[\sqrt {41.7} \] or \[\sqrt {4170 \times {{10}^{ - 2}}} \]
Since we have already determined that
\[\sqrt {4170} \simeq 64\] (Slightly greater than\[64\])
Also,
\[\sqrt {4170 \times {{10}^{ - 2}}} \simeq 64 \times {10^{ - 1}}\]
\[\sqrt {41.7} \simeq 6.4\] Or \[6.4 < \sqrt {41.7} < 6.5\]
Similarly for other we get
\[\sqrt {0.6781} \simeq 0.82\] & \[\sqrt {0.8} \simeq 0.89\]

Note:
> If a natural number m can be expressed as \[{n^2}\], where $n$ is also a natural number, then $m$ is a square number.
> All square numbers end with \[0,1,4,5,6{\text{o}}r9\] 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.