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Factorize the expression $a^4-16b^4$.
The given expression is $a^4-16b^4$.
We have to factorize the expression $a^4-16b^4$.
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.
$(a^2-4b^2)$ 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 $(a^2-4b^2)$.
$(a^2+4b^2)(a^2-4b^2)=(a^2+4b^2)(a+2b)(a-2b)$ [Using (I)]
Hence, the given expression can be factorized as $(a^2+4b^2)(a+2b)(a-2b)$.
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