To find the expanded form of [tex]\(\left(2 - 4ab^2\right)^3\)[/tex], we need to apply the binomial theorem or expand it directly through algebraic manipulation. Here’s the detailed step-by-step process:
First, let’s denote the expression as:
[tex]\[
(x - y)^3 \text{ where } x = 2 \text{ and } y = 4ab^2
\][/tex]
The binomial expansion of [tex]\((x - y)^3\)[/tex] is given by:
[tex]\[
(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
\][/tex]
Now we substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 4ab^2\)[/tex] into the expansion formula:
1. Calculate [tex]\(x^3\)[/tex]:
[tex]\[
x^3 = 2^3 = 8
\][/tex]
2. Calculate [tex]\(3x^2y\)[/tex]:
[tex]\[
3x^2y = 3 \cdot (2^2) \cdot (4ab^2) = 3 \cdot 4 \cdot (4ab^2) = 48ab^2
\][/tex]
3. Calculate [tex]\(3xy^2\)[/tex]:
[tex]\[
3xy^2 = 3 \cdot 2 \cdot (4ab^2)^2 = 3 \cdot 2 \cdot (16a^2b^4) = 96a^2b^4
\][/tex]
4. Calculate [tex]\(y^3\)[/tex]:
[tex]\[
y^3 = (4ab^2)^3 = 64a^3b^6
\][/tex]
Next, we substitute these values back into the expansion formula:
[tex]\[
(2 - 4ab^2)^3 = 2^3 - 3 \cdot 2^2 \cdot (4ab^2) + 3 \cdot 2 \cdot (4ab^2)^2 - (4ab^2)^3
\][/tex]
Combining all the terms, we get:
[tex]\[
8 - 48ab^2 + 96a^2b^4 - 64a^3b^6
\][/tex]
Therefore, the expanded form of [tex]\(\left(2 - 4ab^2\right)^3\)[/tex] is:
[tex]\[
-64a^3b^6 + 96a^2b^4 - 48ab^2 + 8
\][/tex]
