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

Given:

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

To do:

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

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression implies 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.

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

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

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,

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

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

Now,

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

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

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

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

Therefore,

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

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

Updated on: 09-Apr-2023

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