To solve the problem of expanding [tex]\(\left(3a + 8b^3\right)^2\)[/tex], we need to apply the principles of algebraic expansion. This involves using the formula for the square of a binomial:
[tex]\[
(x + y)^2 = x^2 + 2xy + y^2
\][/tex]
In this scenario, we have:
- [tex]\(x = 3a\)[/tex]
- [tex]\(y = 8b^3\)[/tex]
So, we need to expand [tex]\((3a + 8b^3)^2\)[/tex]. Applying the formula gives us:
[tex]\[
(3a + 8b^3)^2 = (3a)^2 + 2(3a)(8b^3) + (8b^3)^2
\][/tex]
Now, let's break this down step by step.
1. Calculate [tex]\((3a)^2\)[/tex]:
[tex]\[
(3a)^2 = 9a^2
\][/tex]
2. Calculate [tex]\(2(3a)(8b^3)\)[/tex]:
[tex]\[
2(3a)(8b^3) = 2 \cdot 3 \cdot 8 \cdot a \cdot b^3 = 48ab^3
\][/tex]
3. Calculate [tex]\((8b^3)^2\)[/tex]:
[tex]\[
(8b^3)^2 = 64b^6
\][/tex]
Finally, combine all the terms we have calculated:
[tex]\[
(3a + 8b^3)^2 = 9a^2 + 48ab^3 + 64b^6
\][/tex]
Therefore, the expanded form of [tex]\((3a + 8b^3)^2\)[/tex] is:
[tex]\[
9a^2 + 48ab^3 + 64b^6
\][/tex]
