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Let's solve the problem step by step.

### Problem Analysis
We are tasked to:
1. Find the coefficient of \(x^2\) in the expansion of \((\frac{1}{x} + 5x)^8\).
2. Find the coefficient of \(x^4\) in the expansion of \((a + 5x)^7\).
3. Set these two coefficients equal and solve for \(a\).

---

### Step 1: Coefficient of \(x^2\) in \((\frac{1}{x} + 5x)^8\)
The general term in the binomial expansion of \((\frac{1}{x} + 5x)^8\) is given by:
\[
T_k = \binom{8}{k} \left(\frac{1}{x}\right)^{8-k} (5x)^k
\]
Simplify:
\[
T_k = \binom{8}{k} \frac{1^{8-k}}{x^{8-k}} (5^k x^k)
\]
\[
T_k = \binom{8}{k} 5^k x^{k - (8-k)} = \binom{8}{k} 5^k x^{2k-8}.
\]

For \(x^2\), set \(2k - 8 = 2\):
\[
2k = 10 \implies k = 5.
\]

Substitute \(k = 5\) into the term:
\[
T_5 = \binom{8}{5} 5^5 x^{2}.
\]

Coefficient of \(x^2\) is:
\[
\binom{8}{5} 5^5.
\]

---

### Step 2: Coefficient of \(x^4\) in \((a + 5x)^7\)
The general term in the binomial expansion of \((a + 5x)^7\) is:
\[
T_k = \binom{7}{k} a^{7-k} (5x)^k.
\]
Simplify:
\[
T_k = \binom{7}{k} a^{7-k} 5^k x^k.
\]

For \(x^4\), set \(k = 4\):
\[
T_4 = \binom{7}{4} a^{7-4} 5^4 x^4.
\]

Coefficient of \(x^4\) is:
\[
\binom{7}{4} a^3 5^4.
\]

---

### Step 3: Equating the Coefficients
Set the coefficients equal:
\[
\binom{8}{5} 5^5 = \binom{7}{4} a^3 5^4.
\]

Simplify:
\[
\frac{8!}{5!(8-5)!} 5^5 = \frac{7!}{4!(7-4)!} a^3 5^4.
\]

Factorials:
\[
\binom{8}{5} = \frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56, \quad \binom{7}{4} = \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1} = 35.
\]

Substitute:
\[
56 \cdot 5^5 = 35 \cdot a^3 \cdot 5^4.
\]

Cancel \(5^4\):
\[
56 \cdot 5 = 35 \cdot a^3.
\]

Simplify:
\[
280 = 35a^3 \implies a^3 = \frac{280}{35} = 8.
\]

Solve for \(a\):
\[
a = \sqrt[3]{8} = 2.
\]

---

### Final Answer
\[
a = 2.
\]

Would you like a detailed breakdown of any part, or have additional questions?

---

### Related Questions
1. What is the general term in a binomial expansion?
2. How do factorials simplify binomial coefficients?
3. Why does the condition \(2k - 8 = 2\) arise when finding the coefficient of \(x^2\)?
4. Can \(a\) have other values in this problem?
5. How would this change if the powers in the expansions were different?

### Tip
When solving problems involving binomial expansions, carefully equate powers and coefficients step by step to avoid errors.

The coefficient of x^2 in the expansion of (1/x + 5x)^8 is equal to the coefficient of x^4 in the expansion of (a + 5x)^7, a ∈ ℝ. Find the value of a.

Learn how to find the value of 'a' in a binomial expansion problem involving coefficients of x^2 and x^4. This step-by-step solution uses the binomial theorem and factorial simplifications, suitable for Grades 11-12.

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