Question

Using Binomial Theorem, evaluate \((102)^5\).

Answer 1

Given:

102⁵

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Step 1: Rewrite using Binomial Theorem

102 = 100 + 2

∴ 102⁵ = (100 + 2)⁵

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Step 2: Apply Binomial Theorem

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

Let a = 100 and b = 2

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Step 3: Substitute values

(100 + 2)⁵ =

100⁵ + 5·100⁴·2 + 10·100³·2² + 10·100²·2³ + 5·100·2⁴ + 2⁵

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Step 4: Simplify

100⁵ = 10,000,000,000
5·100⁴·2 = 1,000,000,000
10·100³·4 = 40,000,000
10·100²·8 = 8,000,000
5·100·16 = 8,000
2⁵ = 32

102⁵ =

10,000,000,000 + 1,000,000,000 + 40,000,000 + 8,000,000 + 8,000 + 32

= 11,048,008,032

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Final Answer:

102⁵ = 11,048,008,032