Question

Which of the following is a perfect square
(a) \[1057\]
(b) \[625\]
(c) \[7928\]
(d) \[6400\]

Answer 1

Hint: Here in this question we have to find which number is perfect square, if we apply the square root to the above numbers then we get a number which is not fraction and decimal then we call it a perfect square. To solve this question we use the division method.

Complete step by step solution:
Squares are the numbers, generated after multiplying a value by itself. Whereas square root of a number is value which on getting multiplied by itself gives the original value.
Now we will check whether the given number is a perfect square one by one.
Now you consider the first number
(a) \[1057\]
The number 1057 we divide it by 7 then quotient will be 151, further we cannot divide by any number. So we are not getting the quotient one. Therefore the number 1057 is not a perfect square.

(b) \[625\]
The number 625, if we divide the number 625 by 5 then the quotient will be 125, again dividing the number 125 by 5 then the quotient will be 25. Again on dividing the number by 5 the quotient will be 5. Again on dividing the number 5 by 5 the quotient will be 1. Here finally we got the quotient 1.
Therefore \[\sqrt {625} = \sqrt {5 \times 5 \times 5 \times 5} \]
And on further simplification
\[ \Rightarrow \sqrt {625} = \sqrt {{5^2} \times {5^2}} \]
Taking the square root for each term
\[ \Rightarrow \sqrt {625} = \sqrt {{5^2}} \times \sqrt {{5^2}} \]
The square and square root will get cancels
\[ \Rightarrow \sqrt {625} = 5 \times 5 = 25\]
Therefore the 625 is a perfect square.

(c) \[7928\]
The number 7928 divided by 8, then the quotient will be 991, the number 991 cannot be divided by any number. Here we will not get the quotient one. Therefore the number 7928 is not a perfect square.

(d) \[6400\]
The number 6400, if we divide the number 6400 by 10 then the quotient will be 640, again dividing the number 640 by 10 then the quotient will be 64. Again on dividing the number by 8 the quotient will be 8. Again on dividing the number 8 by 8 the quotient will be 1. Here finally we got the quotient 1.
Therefore \[\sqrt {6400} = \sqrt {10 \times 10 \times 8 \times 8} \]
And on further simplification
\[ \Rightarrow \sqrt {6400} = \sqrt {{{10}^2} \times {8^2}} \]
Taking the square root for each term
\[ \Rightarrow \sqrt {6400} = \sqrt {{{10}^2}} \times \sqrt {{8^2}} \]
The square and square root will get cancels
\[ \Rightarrow \sqrt {6400} = 10 \times 8 = 80\]
Therefore the 6400 is a perfect square.

Note: The square and the square root are inverse to each other. The number is called the perfect square, when we apply the square root to the number if we obtain the number which is not fraction and decimal. The division method and the tables of multiplication is very important to solve these kinds of problems.