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

Question

UpStudy AI · Accepted Answer

To factor the expression $$ 8a^{3}+36a^{2}+54ab^{2}+27b^{3} $$, we can start by grouping the terms and looking for common factors.

First, we can rearrange the expression:

$$
8a^3 + 36a^2 + 54ab^2 + 27b^3
$$

Next, we can group the first two terms and the last two terms:

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

Now, we can factor out the common factors from each group:

1. From $$ 8a^3 + 36a^2 $$, we can factor out $$ 4a^2 $$:
   $$
   4a^2(2a + 9)
   $$

2. From $$ 54ab^2 + 27b^3 $$, we can factor out $$ 27b^2 $$:
   $$
   27b^2(2a + 1)
   $$

Now, we rewrite the expression with the factored groups:

$$
4a^2(2a + 9) + 27b^2(2a + 1)
$$

Next, we notice that $$ 2a + 9 $$ and $$ 2a + 1 $$ do not have a common factor, so we can try to factor the entire expression as a polynomial.

We can also check if the expression can be factored as a sum of cubes. The expression can be rewritten as:

$$
(2a + 3b)^3
$$

To verify this, we can expand $$ (2a + 3b)^3 $$:

$$
(2a + 3b)(2a + 3b)(2a + 3b)
$$

Using the binomial expansion:

$$
= (2a + 3b)(4a^2 + 12ab + 9b^2)
$$
$$
= 8a^3 + 36a^2b + 54ab^2 + 27b^3
$$

Thus, the original expression can be factored as:

$$
(2a + 3b)^3
$$

So, the final factored form of the expression $$ 8a^{3}+36a^{2}+54ab^{2}+27b^{3} $$ is:

$$
(2a + 3b)^3
$$
$$ (2a + 3b)^3 $$

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