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Factorize 8a^{3}^{ }+ 27b^{3} + 36a^{2}b + 54ab^{2}

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#### Solution

The given expression to be factorized is `8a^3 + 27b^3 + 36a^2 b+ 54ab^2`

This can be written in the form

`8a^3 + 27b^3 + 36a^2b + 54ab^2 = (2a)^3 + (3b)^3 + 36a^2b + 54ab^2`

Take common 18ab from the last two terms,. Then we get

`8a^3 + 27b^3 + 36a^2 b+ 54ab^2 = (2a)^3 +(3b)^3+ 18ab(2a + 3b)`

This can be written in the following form

`8a^3 + 27b^3 + 36a^2 b+ 54ab^2 = (2a)^3 +(3b)^3+ 3.2a.3b(2a + 3b)`

Recall the formula for the cube of the sum of two numbers `(a+b)^3 = a^3 + b^3 + 3ab(a+b)`

Using the above formula, we have `8a^3 + 27b^3 + 36a^2b+ 54ab^2 = (2a +3b)^3`

We cannot further factorize the expression.

So, the required factorization is of `8a^3 + 27b^3 + 36a^2b+ 54ab^2 " is "(2a +3b)^3` .

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