Alright, let's solve the problem step by step.
Question: Find the binomial expansion of [tex]\((2a - 18b)^2\)[/tex]
Solution:
Firstly, we recall the binomial expansion formula, which states:
[tex]\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\][/tex]
In this case, we have [tex]\( (2a - 18b)^2 \)[/tex]. Here, we let [tex]\( x = 2a \)[/tex] and [tex]\( y = -18b \)[/tex], and [tex]\( n = 2 \)[/tex].
Using the binomial expansion:
[tex]\[
(2a - 18b)^2 = \sum_{k=0}^{2} \binom{2}{k} (2a)^{2-k} (-18b)^k
\][/tex]
We will calculate each term of the sum separately.
1. For [tex]\( k = 0 \)[/tex]:
[tex]\[
\binom{2}{0} (2a)^{2-0} (-18b)^0 = 1 \cdot (2a)^2 \cdot 1 = 4a^2
\][/tex]
2. For [tex]\( k = 1 \)[/tex]:
[tex]\[
\binom{2}{1} (2a)^{2-1} (-18b)^1 = 2 \cdot (2a)^1 \cdot (-18b) = 2 \cdot 2a \cdot (-18b) = -72ab
\][/tex]
3. For [tex]\( k = 2 \)[/tex]:
[tex]\[
\binom{2}{2} (2a)^{2-2} (-18b)^2 = 1 \cdot (2a)^0 \cdot (324b^2) = 324b^2
\][/tex]
Combining all these terms, we get the expansion:
[tex]\[
(2a - 18b)^2 = 4a^2 - 72ab + 324b^2
\][/tex]
So the binomial expansion of [tex]\( (2a - 18b)^2 \)[/tex] is:
[tex]\[
\boxed{4a^2 - 72ab + 324b^2}
\][/tex]
