Question

Find the value of $ 54 \times 46 $ using the identity.

Answer 1

Hint:
Start with using the identity $ {a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right) $ and try to express the given product of $ 54 \times 46 $ in form of $ \left( {a - b} \right)\left( {a + b} \right) $ . Then for finding the product now just find the difference of the squares $ {a^2} - {b^2} $ . Notice that transformation $ 54 \times 46 = \left( {50 + 4} \right)\left( {50 - 4} \right) $ is the easy way.

Complete step by step solution:
Here in this problem we are given a product of two numbers, i.e. $ 54 \times 46 $ and using algebraic identity, we need to find the value of this product without actually multiplying \[54\] with $ 46 $ .
The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials. In this way, algebraic identities are used in the computation of algebraic expressions and solving different polynomials.
As we know the algebraic identity $ {a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right) $ , where ‘a’ and ‘b’ can be any number
Now let’s try to express the given product in form of a product of sum and difference of two numbers as given on the right side of the above identity.
 $ \Rightarrow 54 \times 46 = \left( {50 + 4} \right) \times \left( {50 - 4} \right) $ , such that $ 50 $ can be taken as ‘a’ and $ 4 $ is in place of ‘b’.
Therefore, from the above identity, we can say that:
 $ \Rightarrow 54 \times 46 = \left( {50 + 4} \right) \times \left( {50 - 4} \right) = {50^2} - {4^2} $
So, in this way, we expressed the product of two numbers as the difference of two squares.
This can be further solved by using the square of $ 4 $ as $ 16 $ and the square of $ 50 $ can be calculated.
As we know that the square can be distributed in multiplication, i.e. $ {\left( {abc} \right)^n} = {a^n} \times {b^n} \times {c^n} $
 $ \Rightarrow {50^2} = {\left( {5 \times 10} \right)^2} = {5^2} \times {10^2} = 25 \times 100 = 2500 $
Now let’s substitute the squares of both numbers in the above equation:
 $ \Rightarrow 54 \times 46 = \left( {50 + 4} \right) \times \left( {50 - 4} \right) = {50^2} - {4^2} = 2500 - 16 = 2484 $
Therefore, just by using algebraic identities, we calculated the product of two numbers without actually multiplying the two given numbers together.

Note:
Expressing the given product $ 54 \times 46 $ in an easy sum and difference, i.e. $ \left( {50 + 4} \right) \times \left( {50 - 4} \right) $ was the most crucial part of the solution. Notice that any two different numbers can be expressed in such a way, but we don't always get an easier square number. For example, $ 54 \times 44 = \left( {49 + 5} \right)\left( {49 - 5} \right) $ , here we have to find the difference of the square of $ 49 $ , which can be again simplified using an identity $ {49^2} = {\left( {50 - 1} \right)^2} = {50^2} + {1^2} - 2 \times 50 = 2500 + 1 - 100 = 2401 $