Question

Expand the expression \((x+ \frac{1}{x})^6\).

Answer 1

Given:

(x + 1/x)⁶

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Step 1: Use Binomial Theorem

(a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶

Let
a = x,   b = 1/x

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Step 2: Substitute values

(x + 1/x)⁶ =

x⁶ + 6x⁵(1/x) + 15x⁴(1/x²) + 20x³(1/x³) + 15x²(1/x⁴) + 6x(1/x⁵) + 1/x⁶

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Step 3: Simplify

(x + 1/x)⁶ =

x⁶ + 6x⁴ + 15x² + 20 + 15/x² + 6/x⁴ + 1/x⁶

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Final Answer:

(x + 1/x)⁶ = x⁶ + 6x⁴ + 15x² + 20 + 15/x² + 6/x⁴ + 1/x⁶