Question

Evaluate the following using algebraic identities:
$\left( {2x + y} \right)\left( {2x - y} \right)$

Answer 1

Hint: In the given question, we have to evaluate the product of two binomials as given to us in the problem using the algebraic identities. So, we will first find the algebraic identity that resembles the form of the product of both the binomials. Then, we use the algebraic identity to get an expression for the product.

Complete step-by-step solution:
Given question requires us to find the value of the square of the product $\left( {2x + y} \right)\left( {2x - y} \right)$.
We are asked to use the algebraic identities to simplify the product and find it’s value.
So, we have, $\left( {2x + y} \right)\left( {2x - y} \right)$
We can observe that the product of two binomials given to us resembles the form $\left( {a + b} \right)\left( {a - b} \right)$ where $a = 2x$ and $b = y$.
So, we use the algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$. Hence, we get,
$ \Rightarrow {\left( {2x} \right)^2} - {\left( y \right)^2}$
Computing the squares of the terms, we get,
$ \Rightarrow {\left( {2x} \right)^2} - {y^2}$
Using the law of exponents and powers ${\left( {ab} \right)^n} = {a^n}{b^n}$, we get,
$ \Rightarrow {2^2} \times {x^2} - {y^2}$
We know that the value of square number two is $4$. So, we get,
$ \Rightarrow 4{x^2} - {y^2}$
So, the value of $\left( {2x + y} \right)\left( {2x - y} \right)$ calculated using the algebraic identity $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$ is $4{x^2} - {y^2}$.

Note: Before attempting such questions, one should memorize all the algebraic identities and should know their applications in such problems. Care should be taken while carrying out the calculations. We can also verify the answer of the given question by directly multiplying the two binomials using the distributive property and getting the same answer. One must know the laws of exponents and powers to attempt the problem.