SEO Section Header
The square root of a number is another number that when multiplied by itself results in the original number. As mentioned earlier the square root of a number is represented by the symbol, ‘√’ called the radical sign, and the number whose square root is to be found is written inside this symbol. If x is the square root of y then,
\(\sqrt{y}=x\)
As x is the square root of y, x multiplied with itself would generate y as the product, that is,
\(x \times x = y\text{ }or\text{ }x^{2}=y\)
The square root of a number is positive or negative, having the same absolute value. Let us understand this in the context of the square root of 4.
\(2 \times 2 = 4\)
\(\left( -2 \right) \times \left( -2 \right) = 4\)
So the square of –2 is 4, and square of 2 is also 4, and hence, we can say that the square roots of 4 are both 2 and –2.
Hence, \(\sqrt{4} = \pm \text{ }2\)
Therefore, in general \(\sqrt{y} = \pm \text{ }x.\)
Square roots can be calculated by:
- Prime factorization method
- Long division method
Square Root using Long Division Method
The long division method is used to find the square root of any number be it a small or large, or a perfect or an imperfect square, and helps in minimizing the use of calculators. The long division method involves a series of four major steps which are division, multiplication, subtraction and ‘bringing down’ numbers in subsequent steps of the division operation.
The detailed steps are:
Step 1: Pair the digits of the number starting from the ones place and place a bar on each pair. If the total number of digits is odd then place a bar on the first or leftmost digit.
Step 2: Take a divisor whose square is less than or equal to the leftmost digit or digits under the bar. Divide the leftmost digit or digits under the bar by the divisor.
Step 3: Write the quotient and the remainder. Bring down the next pair of numbers under the bar. Write them to the right of the remainder to form the next dividend.
Step 4: For the next divisor, add the previous divisor to its ones place digit and now add a blank (for the ones place digit) on the right of the sum obtained. Now select a ones place digit of the next divisor such that the product of the divisor and the digit is equal to or less than the dividend.
Step 5: Find the remainder.
Repeat above steps until the remainder is zero or less than the divisor.
[Note: Depending on the number of decimal places you want in the square root value, add an equal number of pairs of zero (00) on the extreme right of the number after a decimal point in step 1.]
Let’s find the square root of 2352 up to one decimal place using the long division method.
Step 1: Pair the digits and add the pair 00 on extreme right (as we have to find square root up to one decimal place).
Step 2: Square of 4 is 16, which is less than the first set 23. So 4 is the first divisor. Divide 23 by 4.
Step 3: Write 4 as the quotient and 7 as the remainder. Bring down the next set of numbers, 52.
Step 4: For the next divisor add 4 to 4 which gives 8. Now solve:
8_ × _ < 752 to find _.
88 × 8 = 704 < 752.
So the next divisor is 88.
Step 5: Find the remainder by subtracting 704 from 752 and bring down the next set. The new dividend is 4800.
Step 6: To find the next divisor add 8 to 88, which gives 96.
Solve 96_ × _ < 4800 to find _.
964 × 4 = 3856 < 4800.
So the next divisor is 964.
That’s it!
The required square root is 48.4.
You can increase the decimal places by adding pairs of zeroes on the right of the number.
Solved Examples
Example 1: Find the square root of 5 using the long division method. Answer up to three decimal places.
Solution:
To find the square root of 5 up to three decimal places we will add three sets of ‘00’ on the right of 5. So the number will be 5.000000.
Now apply the steps of the long division method.
So, the square root of 5 is 2.236.
Example 2: Calculate the square root of 18 using the long division method.
Solution:
Let’s find the square root of 18 up to four decimal places by adding four sets of ‘00’ to the right of 18 after a decimal point. So the number is 18.00000000.
Now apply the long division method.
So, the square root of 18 is 4.2426.
The area of a square can be determined by calculating the product of its side lengths. Now suppose the area of a square is given and we want to determine the length of the side of the square. Such a problem can be solved by applying square root as a concept. So we will find the square root of the area, the square root will be the required side length.
The number or expression written inside or under the square root symbol (radical sign) is called a radicand. For example, in the expression, √2, 2 is the radicand.
Numbers whose square root is an integer are called perfect squares, whereas numbers whose square root is a fraction or a decimal are called imperfect squares.