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Chapter 6: Problem 56

Approximate \((0.98)^{8}\) by evaluating the first three terms of \((1-0.02)^{8}\)

Short Answer

Expert verified
The approximate value of \((0.98)^8\) by evaluating first three terms of binomial theorem is 0.84934656

Step by step solution

01

Rewrite the expression

Rewrite the expression in the form \((1 + x)^n\) where \(x\) is a small quantity. In this case, \((0.98)^{8}\) can be rewritten as \((1 - 0.02)^{8}\). This is ideal for the application of the binomial theorem.
02

State the Binomial Theorem

The binomial theorem states that \((1 + x)^n = 1 + nx + \frac{n(n-1) \cdot x^2}{2!} + \ldots\). The theorem allows for the expansion of a binomial raised to a power.
03

Evaluate the First Three Terms

Substitute \(n = 8\) and \(x = -0.02\) into the binomial theorem and evaluate the first three terms. \((1 - 0.02)^8 \approx 1 + 8 \cdot (-0.02) + \frac{8\cdot7 \cdot(-0.02)^2}{2}\). Evaluate this to get an approximate value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expansion
The Binomial Expansion is a powerful algebraic tool. It helps expand expressions in the form of \((1 + x)^n\). This formula allows us to break down complex powers into more manageable parts. Understanding binomial expansion can simplify calculations, especially when dealing with small values of \(x\). For example, in the expression \((1 - 0.02)^8\), the expansion helps us compute each term separately.
Here鈥檚 how the expansion works:
  • \((1 + x)^n = 1 + nx + \frac{n(n-1) \cdot x^2}{2!} + \ldots\)
  • Each term increases in power and involves factorial calculations.
  • This makes it easier to approximate or compare results.
Learning binomial expansion opens new doors to solving complex algebraic problems effectively.
Approximation Techniques
Approximation Techniques are essential in mathematics when dealing with complex computations. They allow us to find approximate values quickly. By using only the first few terms of a series expansion, like the binomial expansion, we can estimate results accurately enough for practical purposes.
In the problem at hand, we use the first three terms of the series to approximate \((0.98)^8\). Here鈥檚 a breakdown:
  • Rewriting as \((1 - 0.02)^8\) makes it ideal for approximation.
  • Using terms: \(1 + 8(-0.02) + \frac{8\cdot7(-0.02)^2}{2}\).
  • Evaluating the terms gives us a close approximation.
These techniques are handy in engineering, physics, and other fields where exact numbers aren't always necessary.
Precalculus Problem Solving
Precalculus Problem Solving involves a lot of strategic thinking. Using concepts like the binomial theorem prepares students for more advanced calculus topics. This skill ensures a solid foundation in algebra and functions.
Here are some tips for effective precalculus problem-solving:
  • Identify patterns and apply known theorems.
  • Break complex problems into simpler parts.
  • Approximate when exact answers are hard to calculate.
In our exercise, recognizing the pattern in \((0.98)^8\) as \((1 - 0.02)^8\) shows mastery in transforming problems. Applying the binomial theorem simplifies and approximates the answer efficiently, showcasing a deep understanding of mathematical practices.

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