Question

How to find Square root of perfect square number?

Answer 1

(i) The square root of a perfect square number can be obtained by finding the prime factorization of the square number, pairing equal factors and picking out one prime factor of each pair. The product of the prime factors thus picked gives the square root of the number.

Note: We may also write the product of prime factors in exponential form and for finding the square root, we take half of the index value of each factor and then multiply.

For example :

196 = \(\underline {2\times2}\) x \(\underline {7\times7}\) = \(\sqrt{196}\) = 2 x 7 = 14 or

196 = 2² x 7₂ \(\Rightarrow\) \(\sqrt{196}\) = \(2^{\frac{2}{2}}\) x \(7^{\frac{2}{2}}\) = 2 x 7 = 14

(ii) \(\sqrt {p\times q}= \sqrt p\,\times\sqrt q\)

(iii) \(\sqrt {\frac{p}{q}}\) = \(\frac{\sqrt p}{\sqrt q}\)

(iv) The square root of a number can also be found by division method. You will be explained this method with the help of an example.

Example: Find the square root of 17424.

Step 1. Take the first pair of digits and find the nearest perfect square. Here 1² = 1.

Step 2. Twice of 1 = 2

Step 3. 2 goes into 7 three times. Put 3 on the top and in the divisor as shown. 23 × 3 = 69.

Step 4. Double 13. You get 26. 26 goes into 52, 2 times. Place 2 on top and in the divisor as shown 2 × 262 = 524.

Step 5. Subtract. The remainder is 0. Therefore, 132 is the exact square root of 17424.

   

Mark off the digits in pairs from right to left

Note: In a decimal number, the pairing of numbers starts from the decimal point. For the integral part it goes from right to left (\(\leftarrow\)) and for the decimal part it goes from left to right, i.e., \(\overset{\leftarrow}{31}\) \(\overset{\leftarrow}{61}\). \(\overset{\rightarrow}{81}\) \(\overset{\rightarrow}{29}\) . The procedure followed is the same as in integral numbers explained above.

(v) If a positive number is not a perfect square, then an approximate value of its square root may be obtained by the division method. \(\sqrt 2\) can be found as :

\(\therefore\) \(\sqrt2\) = 1.414 (approx.)

Also, if n is not a perfect square as 2, then \(\sqrt n\) is not a rational number, e.g.,\(\sqrt 2\) ,\(\sqrt 3\), \(\sqrt 7\) are not rational numbers.