(1000+1)^1/3
(1000(1+1/1000))1/3
(10^3(1+1/1000)^1/3
(10^3(1+0.001))^1/3
10^3*1/3(1+0.001)^1/3
10(1+1/3*0.001)
10(1+0.003)
10*1.003
10.03

Understanding the Binomial Theorem
The Binomial Theorem provides a way to expand expressions of the form (a + b)^n, where n is a non-negative integer. It states that:
(a + b)^n = Σ [nCk * a^(n-k) * b^k] for k = 0 to n
Where nCk represents the binomial coefficient.
Evaluating (1001)^(1/3)
To evaluate (1001)^(1/3) using the Binomial Theorem, we can express 1001 as:
1001 = 1000 + 1
Thus, we need to expand (1000 + 1)^(1/3).
Applying the Binomial Expansion
Using the Binomial Theorem, we have:
(1000 + 1)^(1/3) = Σ [ (1/3)Ck * (1000)^(1/3-k) * (1)^k ] for k = 0 to a few terms
Calculating Initial Terms
- For k = 0:
- (1/3)C0 * (1000)^(1/3) * (1)^0 = 1 * 10 = 10
- For k = 1:
- (1/3)C1 * (1000)^(1/3-1) * (1)^1 = (1/3) * 10^0 * 1 = 1/3
- For k = 2:
- (1/3)C2 * (1000)^(1/3-2) * (1)^2 = (1/3)(-1/6) * 10^(-5/3)
Combining the Results
Adding the results of the first few terms gives:
(1001)^(1/3) ≈ 10 + 1/3 + smaller terms
This indicates that (1001)^(1/3) is approximately 10.33, with smaller terms contributing a minor adjustment.
Conclusion
The Binomial Theorem allows for the approximation of roots and powers, making complex calculations simpler. In this case, (1001)^(1/3) was effectively evaluated through expansion around 1000.