To solve [tex]\((\sqrt{5} + 2)^4\)[/tex], let's break it down step by step.
### Step 1: Expression Understanding
The expression [tex]\((\sqrt{5} + 2)^4\)[/tex] means we are raising the binomial [tex]\((\sqrt{5} + 2)\)[/tex] to the power of 4.
### Step 2: Expand the Binomial Using the Binomial Theorem
The Binomial Theorem states that:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
For our expression, [tex]\(a = \sqrt{5}\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(n = 4\)[/tex]. Therefore, we can expand this using the coefficients given by binomial coefficients:
[tex]\[
(\sqrt{5} + 2)^4 = \sum_{k=0}^{4} \binom{4}{k} (\sqrt{5})^{4-k} (2)^k
\][/tex]
### Step 3: Compute Each Term
1. [tex]\(k = 0\)[/tex]
[tex]\[
\binom{4}{0} (\sqrt{5})^4 (2)^0 = 1 \cdot (\sqrt{5})^4 \cdot 1 = (\sqrt{5})^4
\][/tex]
Since [tex]\((\sqrt{5})^4 = (5^2) = 25\)[/tex]:
[tex]\[
(\sqrt{5})^4 = 25
\][/tex]
2. [tex]\(k = 1\)[/tex]
[tex]\[
\binom{4}{1} (\sqrt{5})^3 (2)^1 = 4 \cdot (\sqrt{5})^3 \cdot 2 = 4 \cdot (\sqrt{5})^3 \cdot 2
\][/tex]
Since [tex]\((\sqrt{5})^3 = 5 \cdot \sqrt{5}\)[/tex]:
[tex]\[
4 \cdot 2 \cdot 5 \cdot \sqrt{5} = 40 \sqrt{5}
\][/tex]
3. [tex]\(k = 2\)[/tex]
[tex]\[
\binom{4}{2} (\sqrt{5})^2 (2)^2 = 6 \cdot (\sqrt{5})^2 \cdot 4 = 6 \cdot 5 \cdot 4 = 120
\][/tex]
4. [tex]\(k = 3\)[/tex]
[tex]\[
\binom{4}{3} (\sqrt{5})^1 (2)^3 = 4 \cdot (\sqrt{5}) \cdot 8 = 32 \sqrt{5}
\][/tex]
5. [tex]\(k = 4\)[/tex]
[tex]\[
\binom{4}{4} (\sqrt{5})^0 (2)^4 = 1 \cdot 1 \cdot 16 = 16
\][/tex]
### Step 4: Sum the Expanded Terms
Now we collect and sum all terms:
[tex]\[
25 + 40\sqrt{5} + 120 + 32\sqrt{5} + 16
\][/tex]
Combine the constants and radical terms separately:
[tex]\[
(25 + 120 + 16) + (40\sqrt{5} + 32\sqrt{5})
\][/tex]
Simplify:
[tex]\[
161 + 72\sqrt{5}
\][/tex]
### Step 5: Numerical Approximation
We need to compute the numerical value.
The numerical result is:
[tex]\[
321.9968943799849
\][/tex]
So, [tex]\((\sqrt{5} + 2)^4 \approx 321.9968943799849\)[/tex].
