To expand [tex]\((1 + (2x)^4)\)[/tex] using the binomial theorem, we start by understanding the binomial theorem itself, which states:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
In this problem, [tex]\(a = 1\)[/tex], [tex]\(b = (2x)^4\)[/tex], and [tex]\(n = 1\)[/tex]. Let's proceed with the expansion:
Given expression:
[tex]\[
\left(1 + (2x)^4\right)^1
\][/tex]
The binomial theorem now simplifies to:
[tex]\[
(1 + b)^1 = 1 + b
\][/tex]
Substituting [tex]\(b = (2x)^4\)[/tex]:
[tex]\[
(1 + (2x)^4)^1 = 1 + (2x)^4
\][/tex]
Now we expand:
[tex]\[
(2x)^4 = 2^4 \cdot x^4 = 16 \cdot x^4 = 16x^4
\][/tex]
So the expanded form becomes:
[tex]\[
1 + 16x^4
\][/tex]
Thus, the expansion of [tex]\((1 + (2x)^4)\)[/tex] is:
[tex]\[
1 + 16x^4
\][/tex]
Therefore, the simplified form is:
[tex]\[
16 x^4 + 1
\][/tex]
