Square Root Tricks for Fast Calculation

Square Root Tricks help find perfect square roots quickly by using place value, last-digit patterns, nearby squares and algebraic identities. This page covers square root formulas, shortcut methods for two-, three- and four-digit numbers, and approximation techniques for numbers that are not perfect squares.

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What Are Square Root Tricks?

Square root tricks are shortcut methods used to calculate or estimate √N without lengthy calculation. For a number N, √N is the non-negative number whose square equals N.

If a² = N, then √N = a. Perfect squares have integer square roots, such as √144 = 12 because 12² = 144. For a non-perfect square, the root lies between two consecutive integers; for example, 7² < 50 < 8², so 7 < √50 < 8.

Square Root Tricks Formula & Tricks

Important Formulas

Basic square root relation
If a² = N, then √N = a

A square root is the number that gives N when multiplied by itself.

Square of a sum
(a + b)² = a² + 2ab + b²

Use this when a number is close to the square of a convenient base.

Square of a difference
(a − b)² = a² − 2ab + b²

This is useful for numbers just below a known square.

Approximation near a perfect square
√(a² + d) ≈ a + d/(2a)

Here d is small compared with a². This gives an estimate, not generally the exact root.

Quick Tricks

Use the last digit to limit the answer

A perfect square can end only in 0, 1, 4, 5, 6 or 9. If the number ends in 1, its square root can end in 1 or 9; if it ends in 4, the root can end in 2 or 8; if it ends in 5, the root ends in 5; if it ends in 6, the root can end in 4 or 6; and if it ends in 9, the root can end in 3 or 7.

Example: For √2116, the root must end in 4 or 6 because 2116 ends in 6.
Pair digits from the right

For a perfect square, separate digits into pairs from the right. The leftmost group gives the higher place value of the root, while the final digit helps identify its unit digit.

Example: In 2116, write 21|16. Since 4² < 21 < 5², the tens digit is 4. The number ends in 6, so test 44 and 46; 46² = 2116.
Check numbers near a base

When a number is close to a known square, write it as (a ± b)² and expand using (a ± b)² = a² ± 2ab + b².

Example: 104² = (100 + 4)² = 10,000 + 800 + 16 = 10,816.
Estimate a non-perfect square from a nearby square

Choose a² close to N and use √N ≈ a + (N − a²)/(2a). The closer a² is to N, the better the estimate.

Example: For √50, use a = 7: √50 ≈ 7 + 1/14 = 7.0714, while the value is approximately 7.0711.

Square Root Tricks Concepts

Last-Digit Rules for Perfect Squares

The unit digit of a perfect square can only be 0, 1, 4, 5, 6 or 9.

The possible unit digits follow from squaring digits 0 to 9. A square ending in 1 may have a root ending in 1 or 9; ending in 4 gives 2 or 8; ending in 5 gives 5; ending in 6 gives 4 or 6; and ending in 9 gives 3 or 7. A perfect square ending in 0 must have a root ending in 0.

Example: 625 ends in 5, so its square root must end in 5. Since 25² = 625, √625 = 25.

Finding the Root of a Four-Digit Perfect Square

For a four-digit square, the first two digits help determine the tens digit, and the last digit helps determine the unit digit.

Separate the number as the first two digits and the last two digits. Find the two consecutive squares between which the first pair lies. This gives the tens digit of the root. Then use the last-digit rule and verify the possible root.

Example: For 4225, 20² = 400 and 30² = 900, so the tens digit is 6 because 42 lies between 6² = 36 and 7² = 49. The ending 25 requires a root ending in 5. Therefore, √4225 = 65.

Square Roots Using Nearby Bases

A number near a convenient base can be squared quickly by using an algebraic identity.

For a base a and difference b, use (a + b)² = a² + 2ab + b² or (a − b)² = a² − 2ab + b². The same identities can be reversed to recognise a perfect square.

Example: For 98², use (100 − 2)² = 100² − 2(100)(2) + 2² = 10,000 − 400 + 4 = 9,604.

Square Root of a Number Near a Perfect Square

If N is close to a², its square root can be estimated by adding approximately (N − a²)/(2a) to a.

The approximation is √N ≈ a + d/(2a), where d = N − a². For a number below a², d is negative. This method gives a decimal estimate and should not be treated as an exact equality unless the result is verified.

Example: For √99, take a = 10 and d = −1. Then √99 ≈ 10 − 1/20 = 9.95; the actual value is approximately 9.9499.

Checking Whether a Number Is a Perfect Square

A number is a perfect square if its square root is an integer.

First check its possible unit digit, then locate it between consecutive known squares. If a candidate root obtained from these checks squares back to the original number, the number is a perfect square.

Example: For 1156, the ending 6 allows a root ending in 4 or 6. Since 34² = 1156, √1156 = 34.

Square Root Tricks Video Lessons

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Square Root Tricks in Vedic Maths

Learn practical Vedic Maths techniques for finding square roots efficiently. This lesson explains shortcut methods and their application to suitable numerical problems.

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Practice Square Root Tricks Questions

Practise published questions related to this topic.

1An article is sold for 40 percent profit. The value of discount given is equal to (5/14) of the marked price. Cost price of the article is how much percent less than its marked price?→ 2After successive discounts of 15% and 10%, a customer pays ₹3,672. What was the marked price?→ 3A five-digit number 7549A is divisible by 4 and 3. If it is divided by 12, what will be the remainder?→ 4A shopkeeper sells one blender for ₹ 1,000 at a gain of 25% and another for ₹ 1,200 at a loss of 20%. Find his overall profit or loss percentage.→ 5Selling price and cost price of an article is Rs. 4000 and Rs. 5000 respectively. What is the loss percentage?→ 6√(√400 + √25) = ?→ 7After allowing a discount of 50% on the marked price of an article, it is sold for ₹61,649. Find its marked price (in ₹).→ 8Simplify: (√98 + √200 - 2√8)²→ 9Find the value of: (√625 - √81) × (√49 + √121)→ 10An article has a marked price of ₹1000.If sold at a 10% discount, what is the selling price?→

Square Root Tricks Quick Quiz

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Quick Revision Notes

Square Root Tricks: Quick Revision

Remember these rules and formulas for rapid calculation.

  • A perfect square can end only in 0, 1, 4, 5, 6 or 9.
  • For a square ending in 1, the root ends in 1 or 9; for 4, it ends in 2 or 8; for 6, it ends in 4 or 6; and for 9, it ends in 3 or 7.
  • Pair digits from the right when analysing large perfect squares.
  • Use (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
  • For N near a², use √N ≈ a + (N − a²)/(2a).
  • Always square the candidate root to verify an exact answer.

Square Root Tricks FAQs

What is the formula for the square of a number near a base?

Use (a + b)² = a² + 2ab + b² or (a − b)² = a² − 2ab + b². For example, 103² = 10,000 + 600 + 9 = 10,609.

Which last digits can a perfect square have?

A perfect square can end only in 0, 1, 4, 5, 6 or 9. It cannot end in 2, 3, 7 or 8.

How can √2116 be found quickly?

Write 21|16. Since 4² < 21 < 5², the tens digit is 4. The final digit 6 gives unit candidates 4 and 6; 46² = 2116, so √2116 = 46.

How do you estimate √50 without a calculator?

Use 7² = 49. With d = 1, √50 ≈ 7 + 1/(2 × 7) = 7.0714. The actual value is approximately 7.0711.

If a perfect square ends in 5, what is the unit digit of its square root?

Its square root must end in 5, because only a number ending in 5 has a square ending in 25. For example, √625 = 25.

How can you check whether 2025 is a perfect square?

The number ends in 25, so its root may end in 5. Since 45² = (40 + 5)² = 1600 + 400 + 25 = 2025, √2025 = 45.

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