Evaluate (1.999)^4

Problem

(1.999)^4

Solution

1. Rewrite the base as a difference of two simpler numbers to make the calculation easier using binomial expansion.

1.999=2−0.001

2. Apply the binomial theorem for (a−b)^4 which follows the pattern a^4−4*a^3*b+6*a^2*b^2−4*a*b^3+b^4

(2−0.001)^4=2^4−4*(2^3)*(0.001)+6*(2^2)*(0.001^2)−4*(2)*(0.001^3)+0.001^4

3. Calculate each term individually.

2^4=16

4*(8)*(0.001)=0.032

6*(4)*(0.000001)=0.000024

4*(2)*(0.000000001)=0.000000008

0.001^4=0.000000000001

4. Combine the calculated terms to find the final sum.

16−0.032+0.000024−0.000000008+0.000000000001

5. Simplify the arithmetic.

16−0.032=15.968

15.968+0.000024=15.968024

15.968024−0.000000008=15.968023992

15.968023992+0.000000000001=15.968023992001

Final Answer

(1.999)^4=15.968023992001

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