Question

Using Binomial theorem estimate 0.99 to the power of 8 up to four decimal places

Answer 1

Answer:

  0.99⁸ ≈ 0.9227

Step-by-step explanation:

The binomial theorem tells us that

   (1 + x)⁸ = 1 + 8x + 28x² + 56x³ + 70x⁴ + 56x⁵ + 28x⁶ + 8x⁷ + x⁸

where the coefficients come from writing down eight rows of Pascal's Triangle.

Since 0.99 = 1 - 0.01, putting x = -0.01 into the above gives

  0.99⁸ = 1 - 8×0.01 + 28×0.0001 - 56×0.000001 + ...

where all the remaining terms are zero in the first 7 decimal places, so too small for us to be concerned with here.  This gives the approximation

  0.99⁸ ≈ 1 - 0.08 + 0.0028 - 0.000056 = 0.922744 ≈ 0.9227

So, to four decimal places, the approximation is

  0.99⁸ ≈ 0.9227

Hope that helps