Methods for finding the square root of a number

Repeated subtraction

Subtract the consecutive odd numbers from $x$ until you are left with zero. The number of times you subtract is the square root of the number.

• Only works for perfect squares

Prime factorization

Represent the number as a product of prime numbers. Divide the number by the smallest prime number until you can’t yield a remainder of $0$, then increase the prime until you can again. The number can then be represented as a product of all prime factors, e.g.

\frac{40}{2}=20 \ \frac{20}{2}=10\ \frac{10}{2}=5\ \frac{5}{5}=1\

here, the prime factorization is $2^3\ast 5$

Then form pairs of factors such that both factors in each pair are equal. Take one factor from each pair and find the product of the factors of each pair.

e.g.

$$144=(2\ast 2)\ast (2\ast 2)\ast (3\ast 3)$$ $$2^2\ast 2^2\ast 3^2$$ $$(2\ast 2\ast 3{)}^2$$ $$=(12{)}^2$$ $$\sqrt{144}=12$$

Estimation

Find the nearest perfect square. e.g. for $\sqrt{15}$, 9 and 16 are the nearest perfect squares, implying $\sqrt{15}$ lives somewhere between 3 and 4.

Long division

Link

1. Place a bar over every pair of digits of the number starting from the right-most side
2. Divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair
3. Bring down the next number, add the last digit of the quotient to the divisor
4. Find a suitable divisor for the new dividend that was carried down
5. Keep doing that
   
   
   

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Related

References