Square and Square Root: Formulas, Tricks and Methods

Square and Square Root explains how a number is multiplied by itself and how its original value is found. This page covers square root formulas, properties of perfect squares, factorisation and long-division methods, simplification rules, estimation, and quick square tricks for numerical questions.

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What Are Square and Square Root?

The square of a number n is n² = n × n. The principal square root of a non-negative number x is the non-negative number √x whose square is x.

For example, 12² = 12 × 12 = 144, so √144 = 12. In the equation y² = 144, both y = 12 and y = −12 are solutions, but √144 denotes only the principal, non-negative root 12. A number with an integer square root is called a perfect square.

Square and Square Root Formula & Tricks

Important Formulas

Square of a number
n² = n × n

Multiply the number by itself. For example, 18² = 18 × 18 = 324.

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

Use this identity when a number is expressed as a convenient base plus a smaller number. For example, 103² = 100² + 2(100)(3) + 3² = 10,609.

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

Use this identity for numbers close to a convenient base. For example, 98² = 100² − 2(100)(2) + 2² = 9,604.

Difference of two squares
a² − b² = (a − b)(a + b)

This factorisation helps calculate or simplify differences of squares. For example, 51² − 49² = (51 − 49)(51 + 49) = 2 × 100 = 200.

Square root of a product
√(ab) = √a × √b, for a ≥ 0 and b ≥ 0

A square factor can be taken outside the root. For example, √72 = √(36 × 2) = 6√2.

Quick Tricks

Squaring a number ending in 5

For a number 10a + 5, multiply a by a + 1 and append 25: (10a + 5)² = a(a + 1) × 100 + 25.

Example: 85²: 8 × 9 = 72, so append 25 to get 7,225.
Squaring numbers near a base

Use (B ± d)² = B² ± 2Bd + d², where B is a convenient base such as 10, 100 or 1,000.

Example: 97² = (100 − 3)² = 10,000 − 600 + 9 = 9,409.
Check the possible last digit

The units digit of a perfect square can only be 0, 1, 4, 5, 6 or 9. Therefore, a number ending in 2, 3, 7 or 8 cannot be a perfect square.

Example: 1,238 is not a perfect square because its units digit is 8.
Use the neighbouring squares for estimation

If a number lies between consecutive perfect squares, its square root lies between the corresponding integers.

Example: √70 lies between √64 = 8 and √81 = 9. Since 70 is closer to 64, √70 is approximately 8.37.

Square and Square Root Concepts

Properties of Squares

The square of a positive or negative number is always non-negative, and opposite numbers have the same square.

For every real number a, (−a)² = a² and a² ≥ 0. The square of an even integer is even, while the square of an odd integer is odd. A perfect square has an integer square root.

Example: (−7)² = 49 and 7² = 49. Also, 16² = 256 is even, while 15² = 225 is odd.

Perfect Squares and Prime Factorisation

A positive integer is a perfect square if every exponent in its prime factorisation is even.

To find the square root using prime factors, group identical prime factors in pairs and take one factor from each pair. If any prime has an odd exponent, the number is not a perfect square.

Example: 1,764 = 2² × 3² × 7², so √1,764 = 2 × 3 × 7 = 42. In 360 = 2³ × 3² × 5, the exponents of 2 and 5 are odd, so 360 is not a perfect square.

Square Root by Prime Factorisation

Prime factorisation gives the exact square root of a perfect square by selecting one factor from each pair.

First write the number as a product of primes. Pair equal prime factors, then multiply one member of every pair. For non-perfect squares, extract the paired factors and leave unpaired factors inside the root.

Example: √2,304 = √(2⁸ × 3²) = 2⁴ × 3 = 48. Also, √180 = √(2² × 3² × 5) = 6√5.

Square Root by the Long-Division Method

The long-division method finds square roots digit by digit by grouping digits into pairs from the decimal point.

For an integer, mark pairs of digits from right to left. Choose the largest square not exceeding the first group, subtract it, bring down the next pair, and choose the next digit so that the product does not exceed the current dividend. The same process continues for decimal pairs.

Example: For √529, the first digit is 2 because 2² ≤ 5. After subtraction and bringing down 29, choose 3 because 23 × 3 = 69 and 529 − 484 = 45; completing the standard process gives √529 = 23.

Simplifying and Estimating Square Roots

Square factors can be removed from under a radical, while a non-perfect square can be estimated between consecutive perfect squares.

For a ≥ 0, √(a²b) = a√b. To estimate √N, locate consecutive integers m and m + 1 such that m² < N < (m + 1)². For fractions, √(a/b) = √a/√b when a and b are non-negative and b is non-zero.

Example: √200 = √(100 × 2) = 10√2. Since 8² = 64 and 9² = 81, √70 lies between 8 and 9.

Square and Square Root Video Lessons

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Practice Square and Square Root Questions

Practise published questions related to this topic.

1What is the square root of 1444?→ 2What is the square root of 4900?→ 3Find the number that is NOT the sum of two different perfect squares.→ 4√625 + √0.25 + √0.0004 = ?→ 5Given that √15 = 3.87 and √150 = 12.24, then find the approximate value of √15 + √1500 + √0.015.→ 6√0.09/0.6 × 0.4/√0.36 × 1.8/√0.36 = ?→ 7√(0.1225/0.25) = ?→ 8If √210.25 = 14.5, then what is the value of √0.00021025?→ 9If x = √225 ÷ √25 + √81 , then what is the value of 2x?→ 10If x = √196 ÷ √49 + √81, then what is the value of 2x?→

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Quick Revision Notes

Square and Square Root Revision Points

Remember these definitions, identities, properties and calculation rules.

  • The square of n is n² = n × n.
  • √x denotes the principal non-negative square root of x.
  • The equation y² = x has two roots, y = ±√x, when x > 0.
  • A perfect square has even exponents in its prime factorisation.
  • The units digit of a perfect square can only be 0, 1, 4, 5, 6 or 9.
  • (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
  • a² − b² = (a − b)(a + b).
  • To simplify a square root, remove factors that occur in pairs under prime factorisation.

Square and Square Root FAQs

What is the difference between √49 and the solutions of x² = 49?

√49 is the principal square root, so √49 = 7. The equation x² = 49 has two solutions: x = 7 and x = −7.

How do you calculate 125² quickly?

Use (100 + 25)² = 100² + 2(100)(25) + 25² = 10,000 + 5,000 + 625 = 15,625.

How can prime factorisation show whether 900 is a perfect square?

900 = 2² × 3² × 5². Every exponent is even, so 900 is a perfect square and √900 = 2 × 3 × 5 = 30.

What is the simplified form of √288?

√288 = √(144 × 2) = 12√2.

Between which two integers does √150 lie?

Since 12² = 144 and 13² = 169, √150 lies between 12 and 13.

What is the square root of 0.0081?

√0.0081 = 0.09 because 0.09² = 0.0081.

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