# Factorize the expression $a^4b^4-16c^4$.

Given:

The given expression is $a^4b^4-16c^4$.

To do:

We have to factorize the expression $a^4b^4-16c^4$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression means writing the expression as a product of two or more factors. Factorization is the reverse of distribution.

An algebraic expression is factored completely when it is written as a product of prime factors.

$a^4b^4-16c^4$ can be written as,

$a^4b^4-16c^4=(a^2b^2)^2-(4c^2)^2$             [Since $16=(4)^2$]

Here, we can observe that the given expression is a difference of two squares. So, by using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize the given expression.

Therefore,

$a^4b^4-16c^4=(a^2b^2)^2-(4c^2)^2$

$a^4b^4-16c^4=[a^2b^2+4c^2][a^2b^2-4c^2]$

Now,

$a^2b^2-4c^2$ can be written as,

$a^2b^2-4c^2=(ab)^2-(2c)^2$                      [Since $4=2^2$]

Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $a^2b^2-4c^2$.

$a^2b^2-4c^2=(ab)^2-(2c)^2$

$a^2b^2-4c^2=(ab+2c)(ab-2c)$.............(I)

Therefore,

$a^4b^4-16c^4=(a^2b^2+4c^2)(ab+2c)(ab-2c)$            [Using (I)]

Hence, the given expression can be factorized as $(a^2b^2+4c^2)(ab+2c)(ab-2c)$.

Updated on: 08-Apr-2023

63 Views 