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