Finding Square Roots
Finding square roots is a core mathematical skill introduced in Class 8 under the chapter Squares and Square Roots. A square root is the inverse operation of squaring a number. While squaring a number means multiplying it by itself (for example, 5 squared = 5 x 5 = 25), finding the square root means determining which number was multiplied by itself to get a given number (the square root of 25 = 5, because 5 x 5 = 25). Square roots appear everywhere in mathematics — from solving equations and working with the Pythagorean theorem to calculating areas and understanding geometric relationships. For perfect square numbers like 4, 9, 16, 25, 36, and so on, the square root is a whole number. But for non-perfect squares like 2, 3, 5, 7, and 10, the square root is an irrational number that goes on forever without repeating. In Class 8, students learn several systematic methods to find square roots: the prime factorisation method (ideal for perfect squares), the long division method (works for any number, including decimals), the repeated subtraction method (conceptual approach using odd numbers), and estimation techniques for non-perfect squares. Each method has its own strengths and is suited to different types of problems. Mastering these methods builds a strong foundation for algebra, coordinate geometry, and trigonometry in higher classes.
What is Finding Square Roots?
The square root of a number is a value that, when multiplied by itself, gives the original number. If x x x = n, then x is the square root of n. The square root of n is written as the square root of n (using the radical symbol) or as n^(1/2).
Key concepts:
Perfect squares: Numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc., are perfect squares because their square roots are whole numbers. For example, the square root of 144 = 12 because 12 x 12 = 144.
Non-perfect squares: Numbers like 2, 3, 5, 6, 7, 8, 10, 11, etc., are not perfect squares. Their square roots are irrational numbers (non-terminating, non-repeating decimals). For example, the square root of 2 is approximately 1.41421356...
Positive and negative square roots: Every positive number has two square roots — one positive and one negative. For example, both 5 and -5 are square roots of 25 because 5 x 5 = 25 and (-5) x (-5) = 25. However, the symbol for the square root of n refers to the positive (principal) square root by convention.
Square root of zero: The square root of 0 is 0, because 0 x 0 = 0.
No real square root of negative numbers: Negative numbers do not have real square roots because no real number multiplied by itself gives a negative result. For example, there is no real number whose square is -9.
Finding Square Roots Formula
Key formulas and properties related to square roots:
1. Basic definition:
If x squared = n, then x = the square root of n (where n is non-negative and x is the positive root).
2. Product property:
The square root of (a x b) = the square root of a multiplied by the square root of b
Example: The square root of 36 = the square root of (4 x 9) = the square root of 4 multiplied by the square root of 9 = 2 x 3 = 6.
3. Quotient property:
The square root of (a / b) = the square root of a divided by the square root of b (where b is not 0)
Example: The square root of (49/16) = the square root of 49 / the square root of 16 = 7/4.
4. Square root of a squared number:
The square root of (a squared) = |a| (the absolute value of a).
5. Perfect squares up to 20:
1 squared = 1, 2 squared = 4, 3 squared = 9, 4 squared = 16, 5 squared = 25, 6 squared = 36, 7 squared = 49, 8 squared = 64, 9 squared = 81, 10 squared = 100, 11 squared = 121, 12 squared = 144, 13 squared = 169, 14 squared = 196, 15 squared = 225, 16 squared = 256, 17 squared = 289, 18 squared = 324, 19 squared = 361, 20 squared = 400.
6. Approximation formula for non-perfect squares:
The square root of n is approximately equal to a + (n - a squared) / (2a), where a squared is the perfect square closest to n. This gives a good first approximation.
Methods
There are four main methods to find the square root of a number:
Method 1: Prime Factorisation Method
This method works best for perfect square numbers. Steps:
1. Find the prime factorisation of the given number.
2. Group the prime factors into pairs of identical factors.
3. Take one factor from each pair.
4. Multiply these factors together to get the square root.
If any prime factor is left unpaired, the number is NOT a perfect square.
Method 2: Long Division Method
This method works for any number — perfect squares, non-perfect squares, and even decimals. Steps:
1. Group the digits of the number into pairs, starting from the units digit and moving left. For decimals, pair digits from the decimal point moving right.
2. Find the largest number whose square is less than or equal to the first group (or first pair).
3. Write this as the first digit of the square root and as the divisor. Subtract and bring down the next pair.
4. Double the current quotient to get the trial divisor. Find the largest digit d such that (trial divisor x 10 + d) x d is less than or equal to the current dividend.
5. Repeat until all pairs have been processed.
Method 3: Repeated Subtraction Method
This is a conceptual method based on the property that every perfect square is the sum of consecutive odd numbers starting from 1.
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
Steps: Subtract consecutive odd numbers (1, 3, 5, 7, 9, ...) from the given number until you reach 0. Count the number of subtractions — that count is the square root. This method is practical only for small perfect squares.
Method 4: Estimation Method
For non-perfect squares, estimate the square root by finding the two consecutive perfect squares between which the number lies.
Example: The square root of 50 lies between the square root of 49 (= 7) and the square root of 64 (= 8). So the square root of 50 is between 7 and 8, closer to 7 since 50 is closer to 49. A more precise estimate can be found using the formula: approximate square root = lower integer + (number - lower perfect square) / (upper perfect square - lower perfect square).
Solved Examples
Example 1: Example 1: Square root by prime factorisation
Problem: Find the square root of 1764 using the prime factorisation method.
Solution:
Step 1: Find the prime factorisation of 1764.
1764 / 2 = 882
882 / 2 = 441
441 / 3 = 147
147 / 3 = 49
49 / 7 = 7
7 / 7 = 1
So, 1764 = 2 x 2 x 3 x 3 x 7 x 7
Step 2: Group into pairs: (2, 2), (3, 3), (7, 7).
Step 3: Take one from each pair: 2, 3, 7.
Step 4: Multiply: 2 x 3 x 7 = 42.
Answer: The square root of 1764 is 42.
Example 2: Example 2: Square root by prime factorisation (larger number)
Problem: Find the square root of 7056 using prime factorisation.
Solution:
Step 1: Prime factorisation of 7056:
7056 / 2 = 3528
3528 / 2 = 1764
1764 / 2 = 882
882 / 2 = 441
441 / 3 = 147
147 / 3 = 49
49 / 7 = 7
7 / 7 = 1
7056 = 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7
Step 2: Pairs: (2, 2), (2, 2), (3, 3), (7, 7). All factors are paired.
Step 3: Take one from each pair: 2, 2, 3, 7.
Step 4: Multiply: 2 x 2 x 3 x 7 = 84.
Answer: The square root of 7056 is 84.
Example 3: Example 3: Square root by long division
Problem: Find the square root of 5929 using the long division method.
Solution:
Step 1: Group into pairs from right: 59 | 29.
Step 2: Find the largest number whose square is less than or equal to 59. Since 7 squared = 49 and 8 squared = 64, we choose 7.
Write 7 as the first digit of the square root. Quotient = 7, Divisor = 7.
7 x 7 = 49. Subtract: 59 - 49 = 10.
Step 3: Bring down the next pair (29). New dividend = 1029.
Step 4: Double the quotient: 2 x 7 = 14. This is the trial divisor (14_).
We need to find d such that (140 + d) x d is less than or equal to 1029.
Try d = 7: 147 x 7 = 1029. Exactly!
Step 5: Quotient = 77, Remainder = 0.
Answer: The square root of 5929 is 77.
Example 4: Example 4: Square root of a decimal by long division
Problem: Find the square root of 2.56 using the long division method.
Solution:
Step 1: Write 2.56. Group: 2 | 56 (for the integer part, just 2; for the decimal part, pair from the decimal point: 56).
Step 2: Find the largest number whose square is less than or equal to 2. Since 1 squared = 1, choose 1.
Quotient = 1, 1 x 1 = 1. Subtract: 2 - 1 = 1.
Step 3: Place the decimal point in the quotient. Bring down 56. New dividend = 156.
Step 4: Double the quotient: 2 x 1 = 2. Trial divisor = 2_.
Find d: (20 + d) x d should be less than or equal to 156.
Try d = 6: 26 x 6 = 156. Exactly!
Step 5: Quotient = 1.6, Remainder = 0.
Answer: The square root of 2.56 is 1.6.
Example 5: Example 5: Repeated subtraction method
Problem: Find the square root of 81 using the repeated subtraction method.
Solution:
Subtract consecutive odd numbers starting from 1:
81 - 1 = 80 (1st subtraction)
80 - 3 = 77 (2nd subtraction)
77 - 5 = 72 (3rd subtraction)
72 - 7 = 65 (4th subtraction)
65 - 9 = 56 (5th subtraction)
56 - 11 = 45 (6th subtraction)
45 - 13 = 32 (7th subtraction)
32 - 15 = 17 (8th subtraction)
17 - 17 = 0 (9th subtraction)
We reached 0 after 9 subtractions.
Answer: The square root of 81 is 9.
Example 6: Example 6: Estimating the square root of a non-perfect square
Problem: Estimate the square root of 75 to one decimal place.
Solution:
Step 1: Find the two perfect squares between which 75 lies.
8 squared = 64 and 9 squared = 81.
So the square root of 75 lies between 8 and 9.
Step 2: Since 75 is closer to 81 than to 64, the square root is closer to 9.
75 - 64 = 11 and 81 - 64 = 17.
Fraction = 11/17 = 0.65 (approximately).
Step 3: Estimated square root = 8 + 0.65 = 8.65 (approximately).
Step 4: Verify: 8.65 x 8.65 = 74.8225 (close to 75).
Try 8.66: 8.66 x 8.66 = 74.9956 (very close).
Answer: The square root of 75 is approximately 8.66.
Example 7: Example 7: Checking if a number is a perfect square
Problem: Is 2352 a perfect square? If not, find the smallest number by which it must be multiplied to make it a perfect square.
Solution:
Step 1: Prime factorisation of 2352:
2352 / 2 = 1176
1176 / 2 = 588
588 / 2 = 294
294 / 2 = 147
147 / 3 = 49
49 / 7 = 7
7 / 7 = 1
2352 = 2 x 2 x 2 x 2 x 3 x 7 x 7
Step 2: Group into pairs: (2, 2), (2, 2), (7, 7). The factor 3 is unpaired.
Step 3: Since 3 is unpaired, 2352 is NOT a perfect square.
Step 4: To make it a perfect square, multiply by 3.
2352 x 3 = 7056 = 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7. All factors now paired.
The square root of 7056 = 2 x 2 x 3 x 7 = 84.
Answer: 2352 is not a perfect square. Multiply by 3 to get 7056, whose square root is 84.
Example 8: Example 8: Square root in a Pythagorean problem
Problem: A right triangle has legs of length 6 cm and 8 cm. Find the hypotenuse using square roots.
Solution:
Step 1: By the Pythagorean theorem:
hypotenuse squared = 6 squared + 8 squared = 36 + 64 = 100
Step 2: hypotenuse = the square root of 100 = 10 cm.
Answer: The hypotenuse is 10 cm.
Example 9: Example 9: Finding the side of a square given its area
Problem: The area of a square field is 6561 sq m. Find the length of each side.
Solution:
Step 1: Area of square = side squared.
So, side = the square root of 6561.
Step 2: Use prime factorisation:
6561 / 3 = 2187
2187 / 3 = 729
729 / 3 = 243
243 / 3 = 81
81 / 3 = 27
27 / 3 = 9
9 / 3 = 3
3 / 3 = 1
6561 = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 3 to the power 8.
Step 3: Pairs: (3,3), (3,3), (3,3), (3,3). Take one from each: 3, 3, 3, 3.
Side = 3 x 3 x 3 x 3 = 81 m.
Answer: Each side of the square field is 81 m.
Example 10: Example 10: Square root of a fraction
Problem: Find the square root of 196/225.
Solution:
Step 1: Use the quotient property.
The square root of (196/225) = the square root of 196 / the square root of 225.
Step 2: Find each square root.
The square root of 196: 196 = 14 x 14, so the square root of 196 = 14.
The square root of 225: 225 = 15 x 15, so the square root of 225 = 15.
Step 3: The square root of (196/225) = 14/15.
Answer: The square root of 196/225 is 14/15.
Real-World Applications
Square roots have extensive practical applications across many fields:
Geometry and Measurement: Square roots are essential for calculating the side of a square when the area is known (side = square root of area). They are also central to the Pythagorean theorem, which is used to find distances, heights, and lengths of diagonals.
Architecture and Construction: Builders use square roots when calculating diagonal measurements of rooms, the length of support beams, and when verifying that corners are square (using the 3-4-5 rule derived from the Pythagorean theorem).
Science and Physics: Many physics formulas involve square roots. For example, the time period of a pendulum is proportional to the square root of its length. The speed of sound and wave calculations also involve square roots.
Finance: The compound interest formula involves taking roots when solving for the number of years or the rate of interest. Standard deviation in statistics (used in financial analysis) requires square roots.
Computer Science: Algorithms for graphics rendering, distance calculations in coordinate systems, and optimisation problems frequently use square root operations.
Navigation and GPS: The distance between two points on a map or between two GPS coordinates is calculated using the distance formula, which involves a square root: distance = the square root of [(x2-x1) squared + (y2-y1) squared].
Key Points to Remember
- The square root of a number n is a value that, when multiplied by itself, gives n.
- Every positive number has two square roots: one positive and one negative. The radical symbol refers to the positive root.
- Perfect squares (1, 4, 9, 16, 25, ...) have whole number square roots.
- The prime factorisation method works by grouping prime factors into pairs and taking one from each pair.
- The long division method works for all numbers, including decimals and non-perfect squares.
- The repeated subtraction method uses the property that perfect squares are sums of consecutive odd numbers.
- The product property states: the square root of (a x b) = the square root of a times the square root of b.
- The quotient property states: the square root of (a/b) = the square root of a divided by the square root of b.
- If a number's prime factorisation has an unpaired factor, it is not a perfect square.
- The square root of a non-perfect square is an irrational number (non-terminating, non-repeating decimal).
Practice Problems
- Find the square root of 3136 using the prime factorisation method.
- Find the square root of 7921 using the long division method.
- Is 4608 a perfect square? If not, find the smallest number by which it must be divided to make it a perfect square.
- Find the square root of 0.0049 using the long division method.
- Estimate the square root of 150 to one decimal place without a calculator.
- The area of a square garden is 10404 sq m. Find the length of each side and the cost of fencing at Rs 20 per metre.
- Find the square root of 11025 by prime factorisation and verify by the long division method.
- A rectangular field has an area of 3600 sq m and its length is 4 times its width. Find the width using square roots.
Frequently Asked Questions
Q1. What is a square root in simple words?
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 36 is 6 because 6 x 6 = 36. Think of it as the reverse of squaring a number.
Q2. What are the different methods to find a square root?
There are four main methods: (1) Prime factorisation — best for perfect squares, group prime factors into pairs. (2) Long division — works for all numbers including decimals. (3) Repeated subtraction — subtract consecutive odd numbers from the number until you reach 0. (4) Estimation — find the two nearest perfect squares and interpolate.
Q3. What is the prime factorisation method for finding square roots?
In this method, you first find the prime factorisation of the number. Then group the prime factors into pairs of identical factors. Take one factor from each pair and multiply them together. The product is the square root. If any factor remains unpaired, the number is not a perfect square.
Q4. What is the long division method for square roots?
The long division method involves grouping the digits into pairs (starting from the units digit), finding the largest digit whose square fits into the first group, doubling the quotient to form a trial divisor, and repeating the process. This method works for any positive number and can give the square root to any desired number of decimal places.
Q5. Can a negative number have a square root?
In the set of real numbers, negative numbers do not have square roots because the square of any real number (positive or negative) is always non-negative. For example, there is no real number whose square is -16. In higher mathematics, imaginary numbers (like i = the square root of -1) are introduced to handle square roots of negative numbers.
Q6. What is the square root of a fraction?
The square root of a fraction a/b equals the square root of a divided by the square root of b (provided b is not zero). For example, the square root of 9/16 = the square root of 9 divided by the square root of 16 = 3/4.
Q7. How do you know if a number is a perfect square?
A number is a perfect square if its prime factorisation has all factors appearing an even number of times (all can be grouped into pairs). Alternatively, if its square root is a whole number, it is a perfect square. Perfect squares always end in 0, 1, 4, 5, 6, or 9 — they never end in 2, 3, 7, or 8.
Q8. What is the square root of 2?
The square root of 2 is approximately 1.41421356. It is an irrational number, meaning its decimal representation goes on forever without repeating. It is one of the most famous irrational numbers and appears frequently in geometry (as the diagonal of a unit square).
Q9. Why do we use square roots in real life?
Square roots are used in many real-life situations: finding the side of a square from its area, calculating distances using the Pythagorean theorem, determining the speed of waves in physics, computing standard deviation in statistics, rendering graphics in computers, and measuring diagonal lengths in construction.
Q10. Is the square root of 0 defined?
Yes, the square root of 0 is 0, because 0 x 0 = 0. Zero is the only number that has exactly one square root (which is 0 itself), unlike positive numbers which have two square roots (one positive and one negative).
