Question

Using Binomial Theorem, evaluate \((101)^4\)

Answer 1

Given:

101⁴

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Step 1: Rewrite using Binomial Theorem

101 = 100 + 1

∴ 101⁴ = (100 + 1)⁴

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Step 2: Apply Binomial Theorem

(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

Let a = 100, b = 1

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Step 3: Substitute values

(100 + 1)⁴ =

100⁴ + 4·100³ + 6·100² + 4·100 + 1

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Step 4: Simplify

100⁴ = 100,000,000
4·100³ = 4,000,000
6·100² = 60,000
4·100 = 400

101⁴ =

100,000,000 + 4,000,000 + 60,000 + 400 + 1

= 104,060,401

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Final Answer:

101⁴ = 104,060,401