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

Given:

The given expression is $a^4-16b^4$.

To do:

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

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.

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

$a^4-16b^4=(a^2)^2-(4b^2)^2$             [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,

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

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

Now,

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

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

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

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

Therefore,

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

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