Question

The value of \[135 \times 135 - 125 \times 125\] is:

Answer 1

Hint: Consider the value of 135 as 'a' and 125 as 'b' and then rewrite the expression as \[{a^2} - {b^2}\]. Therefore, use the identity \[{a^2} - {b^2}\] as (a-b)(a+b) and then substitute it.

Complete step-by-step answer:
In the question, we are given an expression \[135 \times 135 - 125 \times 125\] and we have to find the value of it.
Here, in the question, we have to evaluate \[135 \times 135 - 125 \times 125\] which can be written as \[{135^2} - {125^2}\] as \[135 \times 135\] is also written as \[{135^2}\] and \[125 \times 125\] is also written as \[{125^2}\].
Now, let's consider 135 as 'a' and 125 as 'b'. Thus, we can write \[{135^2} - {125^2}\] as \[{a^2} - {b^2}\].
Here, let's factorize \[\left( {{a^2} - {b^2}} \right)\] first.
We can write the expression \[\left( {{a^2} - {b^2}} \right)\] which can be further written as \[{a^2} - ab + ab - {b^2}\] which can be factorized as .
Thus, we can say that,
\[\left( {{a^2} - {b^2}} \right)\] is equal to (a-b)(a+b)

Now, as we know, that we have considered 135 as 'a' and 125 as 'b' , so, we can write \[{135^2} - {125^2}\] as \[\left( {135 + 125} \right) \times \left( {135 - 125} \right)\] by using the identity \[\left( {{a^2} - {b^2}} \right)\] equals to (a-b)(a+b)
Hence, on evaluating, we can say that \[\left( {135 + 125} \right) \times \left( {135 - 125} \right)\, \Rightarrow \,260 \times 10\, \Rightarrow \,2600\].
Hence, the value of the expression \[135 \times 135 - 125 \times 125\] is 2600.

Note: One can also do the same problem by first finding the value of \[135 \times 135\,\,{\rm{and}}\,\,125 \times 125\] independently and then subtracting latter value from former one to get the desired result. So we can compute \[{135^2}\,as\,\,18225\] and \[{125^2}\,\,as\,\,15625\]. Then subtracting them, we have \[18225 - 15625 = 2600\]. But computing the square of big numbers is time consuming, so we must avoid this method.