Evaluating 1.075 Using the Binomial Theorem: A Comprehensive Guide

When it comes to evaluating expressions such as 1.07^5, modern calculators often provide the easiest solution. However, understanding alternative methods, such as the binomial theorem, can offer insight into approximation techniques. This article will explore how to evaluate 1.07^5 using the binomial theorem, discuss the reasoning behind this approach, and provide an in-depth analysis of the results.

Using the Binomial Theorem for Approximation

The binomial theorem is particularly useful when an exact calculation is not necessary, and an approximate solution is sufficient. It allows us to express the power of a binomial as a sum of terms. For the case of 1.07^5, the binomial theorem states:

(a b)^n a^n nC_1 a^{n-1}b nC_2 a^{n-2}b^2 ... nC_{n-1} ab^{n-1} b^n

1.07^5 1^5 5C_1 1^4 cdot 0.07 5C_2 1^3 cdot 0.07^2 5C_3 1^2 cdot 0.07^3 5C_4 1 cdot 0.07^4 0.07^5

Step-by-Step Calculation

Let's break down the calculation step-by-step:

1^5 1 5C_1 1^4 cdot 0.07 5 cdot 0.07 0.35 5C_2 1^3 cdot 0.07^2 10 cdot 0.0049 0.049 5C_3 1^2 cdot 0.07^3 10 cdot 0.000343 0.00343 5C_4 1 cdot 0.07^4 5 cdot 0.00002401 0.00012005 0.07^5 0.0000016807

Adding these terms together:

1 0.35 0.049 0.00343 0.00012005 0.0000016807 1.4025517307

Comparison with Direct Calculation

For comparison, we can also calculate 1.07^5 directly through successive multiplications:

1.07 cdot 1.07 1.1449 1.1449 cdot 1.07 1.225043 1.225043 cdot 1.07 1.31079601 1.31079601 cdot 1.07 1.4025517307

To get a decent approximation, we can use the first two or three terms of the binomial expansion:

1 5 cdot 1 cdot 0.07 1.35 Adding the next term, we get: 1.35 10 cdot 1 cdot 0.07^2 1.399

Conclusion

Using the binomial theorem to evaluate 1.07^5 provides a method to find an approximate solution without needing a calculator. While direct computation through successive multiplications is straightforward, the binomial theorem offers a theoretical understanding of how powers work and can be useful in more complex scenarios.

Key Takeaways:

The binomial theorem is a powerful tool for approximating powers. The first two terms can provide a good approximation. Successive multiplications are also an accurate method.

Understanding these methods can enhance your problem-solving skills and provide additional insight into mathematical concepts.

Frequently Asked Questions

Q1: What is the binomial theorem?

The binomial theorem is a formula for expanding powers of binomials. It provides a way to express the power of a binomial as a sum of terms, simplifying the calculation.

Q2: Why is the binomial theorem useful for approximation?

The binomial theorem allows us to approximate powers by using only a few terms, making it useful when a precise calculator is not available or when an approximate solution is sufficient.

Q3: How does the binomial theorem compare to direct calculation?

Both methods are accurate, but the binomial theorem provides a theoretical understanding and can be used in more complex scenarios, while direct calculation through successive multiplications is straightforward and precise.