Factorize the expression $a^4-\frac{1}{b^4}$.

Given:

The given algebraic expression is $a^4-\frac{1}{b^4}$.

To do:

We have to factorize the expression $a^4-\frac{1}{b^4}$.

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.

$a^4-\frac{1}{b^4}$ can be written as,

$a^4-\frac{1}{b^4}=(a^2)^2-(\frac{1}{b^2})^2$

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,

$a^4-\frac{1}{b^4}=(a^2)^2-(\frac{1}{b^2})^2$

$a^4-\frac{1}{b^4}=(a^2+\frac{1}{b^2})(a^2-\frac{1}{b^2})$

Now,

$(a^2-\frac{1}{b^2})$ can be written as,

$(a^2-\frac{1}{b^2})=a^2-(\frac{1}{b})^2$

Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $(a^2-(\frac{1}{b})^2)$.

$a^2-(\frac{1}{b})^2=(a+\frac{1}{b})(a-\frac{1}{b})$.............(I)

Therefore,

$a^4-\frac{1}{b^4}=(a^2+\frac{1}{b^2})(a+\frac{1}{b})(a-\frac{1}{b})$                [Using (I)]

Hence, the given expression can be factorized as $(a^2+\frac{1}{b^2})(a+\frac{1}{b})(a-\frac{1}{b})$.