Question

Find the smallest whole number by which 2800 should be multiplied so as to get a perfect square number. Also, find its square root.

Answer 1

Hint: We use the definition of perfect square and write the prime factorization of the given number. Make pairs or squares of numbers in the prime factorization and check which value has to be multiplied to complete the remaining square. Calculate the value and find its square root by cancelling square root by square power.
* A number is a perfect square if it’s under root or square root is a whole number.
* Prime factorization is a process of writing a number in multiple of its factors where all factors are prime numbers.

Complete answer:
We first write the prime factorization of the number 2800
\[2800 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 7\]
Now we make pairs of two each where the numbers in the pairs are same
\[ \Rightarrow 2800 = \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {5 \times 5} \right) \times 7\]
Use the law of exponents that states that we can collect the powers when base are same
\[ \Rightarrow 2800 = {2^2} \times {2^2} \times {5^2} \times 7\]
To male this number a perfect square we have to complete the square of 7 i.e. we have to multiply by 7
\[ \Rightarrow 2800 \times 7 = {2^2} \times {2^2} \times {5^2} \times 7 \times 7\]
\[ \Rightarrow 19600 = {2^2} \times {2^2} \times {5^2} \times {7^2}\]
Since all the factors in the right hand side are square of prime factors, then the number 19600 is a perfect square.
Now we calculate its square root.
We can write the prime factorization of 19600 by pairing the factors using the law of exponents that states we can combine the power when each number has the same power.
\[ \Rightarrow 19600 = {\left( {2 \times 2 \times 5 \times 7} \right)^2}\]
Calculate the product
\[ \Rightarrow 19600 = {\left( {140} \right)^2}\]
Take square root on both sides of the equation
\[ \Rightarrow \sqrt {19600} = \sqrt {{{\left( {140} \right)}^2}} \]
Since we know square root means the power \[\dfrac{1}{2}\]
\[ \Rightarrow \sqrt {19600} = {\left[ {{{\left( {140} \right)}^2}} \right]^{\dfrac{1}{2}}}\]
Multiply the terms in the power
\[ \Rightarrow \sqrt {19600} = {\left( {140} \right)^{2 \times \dfrac{1}{2}}}\]
\[ \Rightarrow \sqrt {19600} = 140\]

\[\therefore \]The smallest whole number which should be multiplied to 2800 to make it a perfect square is 7 and the square root of the perfect square 19600 is 140.

Note:
Many students have an approach of multiplying the number 2800 by 1, then by 2, then by 3 and so on and then they check if each number obtained is a perfect square or not. This is a very long process and we have to repeat the steps again for each number which is also very lengthy. Students should avoid this procedure and use the concept of prime factorization.