Key Facts on Square number with short-cuts to find Square root of a number

Key Facts on Square number with short-cuts to find Square root of a number

When a perfect square is written as a product of its prime factors each prime factors will appear an even number of times. E.g.
7016 = 2 × 2 × 2 × 2 × 21 × 21.

[Prime factors 2 and 21 are occurs even number of times. 2 has appeared four times and 21 has appeared two times.]

The difference between the square of two consecutive natural numbers is always equal to the sum of the natural numbers. Thus,
41² - 40² = (40 + 41) = 81.
45² - 44² = (44 + 45) = 89.

This is very useful when used in opposite direction- i.e. given that the difference between the square of two consecutive integers is 91, you should realize that the numbers should be 45 and 46.

• The value of a perfect square has to end in 1, 4, 5, 6, 9 or an even number of zeros.
• A perfect square cannot end in 2, 3, 7 or 8 or an odd number of zeros.
• The sum of squares of the first ‘n’ natural numbers is given by,
$$\frac{{\left( {\text{n}} \right)\left( {{\text{n}} + 1} \right)\left( {2{\text{n}} + 1} \right)}}{6}$$
• The square of a number is always non- negative.

Square Root:
Finding square root of a number:
Let number is 7016.

Step 1: Write down the number 7016 as product of its prime factors.
7016 = 2 × 2 × 2 × 2 × 21 × 21.
7016 = 2⁴ × 21²

Step 2: the required square root is obtained by halving the values of the power. Hence,
$$\sqrt {7016} $$  = 2² × 21.

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