Question 9 of 20intermediate🔧 ApplyNumerical4 marks

Using binomial theorem, evaluate each of the following: (99)^5

Correct Answer

9509900499

Exercise: EXERCISE 7.1 | Q: 9 | (Page 8)
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Explanation

This question follows the same pattern as Example 2 in the textbook which computes (98)^5. The student should express 99 as (100 - 1) and apply the binomial theorem. The key is to choose numbers close to powers of 10 to make calculations manageable.

Solution Steps

  1. Step 1: Express 99 as the difference of two numbers with easier powers. We write 99 = 100 - 1 Therefore, (99)^5 = (100 - 1)^5

  2. Step 2: Apply the Binomial Theorem expansion. (100 - 1)^5 = ⁵C₀(100)⁵ - ⁵C₁(100)⁴(1) + ⁵C₂(100)³(1)² - ⁵C₃(100)²(1)³ + ⁵C₄(100)(1)⁴ - ⁵C₅(1)⁵

  3. Step 3: Substitute the values of binomial coefficients. = 1×(100)⁵ - 5×(100)⁴ + 10×(100)³ - 10×(100)² + 5×(100) - 1

  4. Step 4: Calculate each term. = 10000000000 - 5×1000000005 \times 100000000 + 10×100000010 \times 1000000 - 10×1000010 \times 10000 + 5×1005 \times 100 - 1 = 10000000000 - 500000000 + 10000000 - 100000 + 500 - 1

  5. Step 5: Simplify to get the final answer. = 10000000000 + 10000000 + 500 - 500000000 - 100000 - 1 = 10010000500 - 500100001 = 9509900499

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