Square Of a Number–
The square of a number is that number raised to the power 2. Thus, if ‘a’ is a number, then the square of a is written as \({ a }^{ 2 } \) and is given by
\({ a }^{ 2 } \) = a x a
That is, the square of a number is obtained by multiplying it by itself.
If a x a = b i.e. \({ a }^{ 2 } \) = b, then we say that the square of number a is number b or the number b is the square of number a.
For example,
\({ 2 }^{ 2 } \) = 2 x 2 = 4, so we say that the square of 2 is 4
\({ 3 }^{ 2 } \) =3 x 3 = 9, so we say that the square of 3 is 9
\({ 12 }^{ 2 } \)= 12 x 12 = 144, so we say that the square of 12 is 144 etc.
Perfect Square Or Square Number –
A natural number n is called a perfect square or a square number if there exists a natural number m such that n = \({ m }^{ 2 } \)
In other words, a natural number is a perfect square if it is the square of some natural number.
Since, 4 = 2 x 2 = \({ 2 }^{ 2 } \). Therefore, 4 is a square number:
Also,
25 =5×5 = \({ 5 }^{ 2 } \), so 25 is a square number or a perfect square.
144 = 12 x 12 = \({ 12 }^{ 2 } \), so 144 is a square number or a perfect square etc.
In order to check whether a given natural number is a perfect square or not, we follow the
following steps: –
Step I – Obtain the natural number.
Step II – Write the number as a product of prime factors.
Step III – Group the factors in pairs in such a way that both the factors in each pair are equal.
Step IV – See whether some factor is left over or not. If no factor is left over in grouping, then the given number is a perfect square. Otherwise, it is not a perfect square.
Step V – To obtain the number whose square is the given number take one factor from each group and multiply them.
Following examples will illustrate the above procedure:
Prime Factors of 144 are 2 x 2 x 2 x 2 x 3 x 3. About Us Disclaimer Privacy Policy Contact Us
Illustrative Examples :
Example 1 : Is 225 a perfect square? if so, find the number whose square is 225.
Solution– Resolving 225 into prime factors, we obtain
225 =3x3x5x5
Grouping the factors in pairs in such a way that both the factors in each pair are equal, we have
225 = (3 x 3) x (5 x 5)
Clearly, 225 can be grouped into pairs of equal factors and no factor is left over. Hence, 225 is a perfect square. Again, 225=(3×5)x(3×5)
=15×15= \({ 15 }^{ 2 } \)
So, 225 is the square of 15.
Example 2 : Show that 63504 is a perfect square. Also, find the number whose square is 63504.
Solution – Resolving 63504 into prime factors, we obtain
63504 =2x2x2x2x3x3x3x3x7x7
Grouping the factors in pairs of equal factors, we obtain
63504 = (2 x 2) x (2 x 2) x (3 x 3) x (3 x 3) x (7 x 7)
Clearly, no factor is left over in grouping the factors in pairs of equal factors, So, 63504 is a perfect square. Again,
63504 =(2x2x3x3x7)x(2x2x3x3x7) [Grouping first factors in each group]
= 252 x252= \({ 252 }^{ 2 } \)
Thus, 63504 is the square of 252.
Example 3 : Show that 17640 is not a perfect square.
Solution– Resolving 17640 into prime factors, we have
17640 =2x2x2x3x3x5x7x7
Grouping the factors into pairs of equal factors, we get
17640 = (2×2)x(3×3)x(7×7)x(2×5)
Clearly, on grouping into pairs of equal factors,
we are left with two factors 2 and 5 which cannot be paired.
Hence,17640 is not a perfect square.