Question

Expand the expression \((\frac{x}{3} + \frac{1}{x})^5\).

Answer 1

Given:

( x/3 + 1/x )⁵

---

Step 1: Apply Binomial Theorem

(a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Let
a = x/3,   b = 1/x

---

Step 2: Evaluate each term

a⁵ = x⁵/243

5a⁴b = 5x³/81

10a³b² = 10x/27

10a²b³ = 10/(9x)

5ab⁴ = 5/(3x³)

b⁵ = 1/x⁵

---

Step 3: Write the expansion

(x/3 + 1/x)⁵ =

x⁵/243 + 5x³/81 + 10x/27 + 10/(9x) + 5/(3x³) + 1/x⁵

---

Final Answer:

(x/3 + 1/x)⁵ = x⁵/243 + 5x³/81 + 10x/27 + 10/(9x) + 5/(3x³) + 1/x⁵