Question

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

Answer 1

Hint:
Here we have to find the value of a given product using the algebraic identity. First, we will convert the first number into a sum of two numbers and then we will convert the second number into a difference of two numbers. Then we will use the algebraic identity to solve the product of sum and difference of two numbers. From there, we will get the final value of a given expression or given product of two numbers.

Complete step by step solution:
The given expression is the product of two numbers i.e.
 $54\times 46$.
We need to calculate its value using algebraic identity.
We will convert the first number i.e. 54 into a sum of two numbers.
We can write 54 as the sum of 50 and 4.
We will convert the second number i.e. 46 into a difference of two numbers.
We can write 46 as differences of 50 and 4.
Thus, $54\times 46$ becomes;
$=\left( 50+4 \right)\left( 50-4 \right)$
We know from the algebraic identities that
$\Rightarrow \left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ …….. $\left( 1 \right)$
Therefore, we have the values:-
$\begin{align}
  & a=50 \\
 & b=4 \\
\end{align}$
Now, we will substitute the value of $a$ and $b$ in equation 1.
$\Rightarrow \left( 50+4 \right)\left( 50-4 \right)={{50}^{2}}-{{4}^{2}}$
Substituting the value of squares of two numbers, we get
$\Rightarrow \left( 50+4 \right)\left( 50-4 \right)=2500-16$
On subtracting the numbers on left hand side of this equation, we get
$\Rightarrow \left( 50+4 \right)\left( 50-4 \right)=2486$

Thus, the value of $54\times 46$ using algebraic identity is 2486.

Note:
Since we have solved this question using algebraic identities. Let’s define it to understand it deeply. The algebraic identities are the equations which are true for all values of variables in them. Algebraic identities are also used for the factorization of different types of polynomials. In this way, they are used in the computation of algebraic expressions and solving different polynomials. Remember that all the algebraic identities are derived from the binomial theorem.