Question

Using Binomial Theorem, evaluate \((99)^5\).

Answer 1

Given:

99⁵

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Step 1: Rewrite using Binomial Theorem

99 = 100 − 1

∴ 99⁵ = (100 − 1)⁵

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Step 2: Apply Binomial Theorem

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

Let a = 100, b = 1

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Step 3: Substitute values

(100 − 1)⁵ =

100⁵ − 5·100⁴ + 10·100³ − 10·100² + 5·100 − 1

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Step 4: Simplify

100⁵ = 10,000,000,000
5·100⁴ = 500,000,000
10·100³ = 10,000,000
10·100² = 100,000
5·100 = 500

99⁵ =

10,000,000,000 − 500,000,000 + 10,000,000 − 100,000 + 500 − 1

= 9,509,900,499

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Final Answer:

99⁵ = 9,509,900,499