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Factorize the expression $75a^3b^2-108ab^4$.
Given:
The given expression is $75a^3b^2-108ab^4$.
To do:
We have to factorize the expression $75a^3b^2-108ab^4$.
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.
$75a^3b^2-108ab^4$ can be written as,
$75a^3b^2-108ab^4=3ab^2(25a^2-36b^2)$ (Taking $3ab^2$ common)
$75a^3b^2-108ab^4=3ab^2[(5a)^2-(6b)^2]$ [Since $25=5^2, 36=6^2$]
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,
$75a^3b^2-108ab^4=3ab^2[(5a)^2-(6b)^2]$
$75a^3b^2-108ab^4=3ab^2(5a+6b)(5a-6b)$
Hence, the given expression can be factorized as $3ab^2(5a+6b)(5a-6b)$.
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