Computing 1.02^4 Using the Binomial Theorem: A Comprehensive Guide

Introduction

Understanding and applying the binomial theorem is a valuable skill in advanced mathematics, particularly in problems involving exponents. In this article, we will explore how to compute the value of 1.02^4 using the binomial theorem, rounding the result to two decimal places. This method provides a detailed and accurate approach to solving such mathematical problems.

What is the Binomial Theorem?

The binomial theorem is a powerful tool for expanding expressions of the form (a x)^n. It states that:

(a x)^n sum_{k0}^{n} binom{n}{k} a^{n-k} x^k

This theorem allows us to expand binomials quickly and accurately, which is particularly useful in various mathematical and computational contexts.

Computing 1.02^4 Using the Binomial Theorem

To compute 1.02^4, we will apply the binomial theorem with x 0.02 and n 4. This allows us to express the expression as:

binom{4}{0} 0.02^0 binom{4}{1} 0.02^1 binom{4}{2} 0.02^2 binom{4}{3} 0.02^3 binom{4}{4} 0.02^4

Step-by-Step Calculation

Let's break down the computation step by step:

Term for k 0

binom{4}{0} 0.02^0 1

Term for k 1

binom{4}{1} 0.02^1 4 cdot 0.02 0.08

Term for k 2

binom{4}{2} 0.02^2 6 cdot 0.0004 0.0024

Term for k 3

binom{4}{3} 0.02^3 4 cdot 0.000008 0.000032

Term for k 4

binom{4}{4} 0.02^4 1 cdot 0.00000016 0.00000016

Now, let's sum up these terms:

1.02^4 approx 1 0.08 0.0024 0.000032 0.00000016

Summing the Terms

Adding the terms together step by step:

1 0.08 1.08 1.08 0.0024 1.0824 1.0824 0.000032 approx 1.082432 1.082432 0.00000016 approx 1.08243216

Rounding the result to two decimal places:

1.08243216 approx 1.08

Conclusion

The value of 1.02^4 computed using the binomial theorem, rounded to two decimal places, is approximately 1.08. This method provides an accurate and detailed approach to solving similar mathematical problems.

Additional Examples

Let's consider another example using the binomial theorem:

Example: 1.01^3

We can apply a similar approach with x 0.01 and n 3. Using the binomial theorem:

binom{3}{0} 0.01^0 binom{3}{1} 0.01^1 binom{3}{2} 0.01^2 binom{3}{3} 0.01^3

Computing each term:

Term for k 0

binom{3}{0} 0.01^0 1

Term for k 1

binom{3}{1} 0.01^1 3 cdot 0.01 0.03

Term for k 2

binom{3}{2} 0.01^2 3 cdot 0.0001 0.0003

Term for k 3

binom{3}{3} 0.01^3 1 cdot 0.000001 0.000001

Summing the terms:

1.01^3 approx 1 0.03 0.0003 0.000001 1.030301 approx 1.03

This method can be applied to a wide range of mathematical problems involving exponents and binomials.

Conclusion

The binomial theorem is a powerful tool for expanding and solving expressions involving exponents. By understanding and applying this theorem, you can accurately compute values such as 1.02^4 and 1.01^3, providing a solid foundation in advanced mathematical techniques.