Question 7 of 20intermediate🔧 ApplyNumerical4 marks

Using binomial theorem, evaluate each of the following: (102)^5

Correct Answer

11040808032

Exercise: EXERCISE 7.1 | Q: 7 | (Page 8)
For More Understanding

Explanation

Following the same method as Example 2 in the textbook which computes (98)^5 by expressing 98 as (100-2), we express 102 as (100+2). The binomial expansion is then applied with all positive terms since we are adding. The binomial coefficients used are from the standard formula: ^5C_0=1, ^5C_1=5, ^5C_2=10, ^5C_3=10, ^5C_4=5, ^5C_5=1.

Solution Steps

  1. Step 1: Express 102 as sum of two numbers: 102 = 100 + 2

  2. Step 2: Apply binomial theorem: (102)^5 = (100 + 2)^5 = ^5C_0(100)^5 + ^5C_1(100)^4(2) + ^5C_2(100)^3(2)^2 + ^5C_3(100)^2(2)^3 + ^5C_4(100)(2)^4 + ^5C_5(2)^5

  3. Step 3: Substitute binomial coefficients: = 1(100)^5 + 5(100)^4(2) + 10(100)^3(4) + 10(100)^2(8) + 5(100)(16) + 1(32)

  4. Step 4: Calculate each term: = 10000000000 + 5(100000000)(2) + 10(1000000)(4) + 10(10000)(8) + 5(100)(16) + 32

  5. Step 5: Simplify: = 10000000000 + 1000000000 + 40000000 + 800000 + 8000 + 32

  6. Step 6: Add all terms: = 11040808032

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