How to Find Square Root of Numbers and Fractions

In this article, you will learn how to calculate the square root of numbers, fractions using various methods. But we would like to start by the definitions of square and square root –

Square

A number multiplied by itself is known as the square of the given number. For example, square of 6 is $6\times6=36$.

Square Root

Square root of a given number is that number which when multiplied by itself is equal to the given number.
For example, the square root of 81 is 9 because of $9^{2}=9\times9=81$.
The square root of a number is denoted by the symbol $\sqrt{}$ ,called the radical sign. Thus, $\sqrt{81}=9$, $\sqrt{64}=8$ and so on. Note that $\sqrt{1}=1$.

Methods of Finding the Square Root

I. Prime Factorization Method

  1. Find the prime factors of the given number.
  2. Group the factors in pairs.
  3. Take one number from each pair of factors and then multiply them together.

This product is the square root of the given number.
Example: Find the square root of 207025.

Solution: \[ 207025=\underbrace{5\times5}\times\underbrace{7\times7}\times\underbrace{13\times13} \]
Hence, $\sqrt{207025}=5\times7\times13=455$.
This method is used when the given number is a perfect square or when every prime factor of that number is repeated twice.

II. Method of Division

This method is used when the number is large and the factors cannot be easily determined.
The working rule is explained with the help of the following example:

  • Step 1: The digits of a number, whose square root is required, are separated into periods of two beginning from the right. The last period may be either single digit or a pair.

square root

  • Step 2: Find a number (here, 4) whose square may be equal to or less than the first period (here, 22).
  • Step 3: Find out the remainder (here, 6 and bring down the next period (here, 65).
  • Step 4: Double the quotient (here, 4) and write to the left (here, 8).
  • Step 5: The divisor of this stage will be equal to the above sum (here, 8) with the quotient of this stage (here, 7) suffixed to it (here, 87).
  • Step 6: Repeat this process (step 4 and step 5) till all the periods get exhausted.
    The quotient (here, 476) is equal to the square root of the given number (here, 226576)

Calculate Square Root of a Decimal

In this case, two type of situation may arise. We understand this with examples.

  • Case 1: If a decimal has an even number of decimal places (say 12.1801), then it can be written as $\frac{121801}{10000}$. Therefore, $\sqrt{12.1801}=\frac{\sqrt{121801}}{\sqrt{10000}}$. Now, $\sqrt{10000}=100$ and $\sqrt{121801}$ can be found using above methods and its equal to 349. Therefore, $\sqrt{12.1801}=\frac{349}{100}=3.49$.
  • Case 2: If a decimal has an odd number of decimal places (say 4.9), then it can be written as $\frac{490}{100}$ (note that we have written it so that the square root of the denominator can be easily found). Therefore $\sqrt{4.9}=\frac{\sqrt{490}}{\sqrt{100}}$. Now $\sqrt{100}=10$ and $\sqrt{490}$ can be found by above method and its value will be approximately 22. Therefore, $\sqrt{4.9}=\frac{22}{10}=2.2$.
    Note: For better approximation write $4.9$ as $\frac{49000}{10000}$ and then determine the square root as the above method. For more better approximation, add some more pair of zeros in both numerator and denominator and then take the square root.

Calculate Square Root of a Fraction

  1. If the denominator is a perfect square: The square root is found by taking the square root of the numerator and denominator separately.
    For example,

    1. $\sqrt{\frac{2704}{49}}=\frac{\sqrt{2704}}{\sqrt{49}}=\frac{\sqrt{52\times52}}{\sqrt{7\times7}}=\frac{52}{7}$
    2. $\sqrt{\frac{44}{25}}=\frac{\sqrt{44}}{\sqrt{25}}=\frac{\sqrt{44}}{\sqrt{5\times5}}=\frac{6.6332}{5}\approx1.3266$
  2. If the denominator is not a perfect square: The fraction is converted into decimal and then square root is obtained or the denominator is made perfect square by multiplying and dividing by a suitable number and then its square root is obtained.
    For example,

    1. $\sqrt{\frac{354}{43}}=\sqrt{8.2325}\approx2.8692$
    2. $\sqrt{\frac{461}{32}}=\sqrt{\frac{461\times2}{32\times2}}=\frac{\sqrt{922}}{\sqrt{64}}=\frac{30.3644}{8}\approx3.7955$

 

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