Problem 71 - Calculus with Concepts in Calculus

Question

Modify the expression, and then find its approximate value by calculator or computer. $$ \frac{69^{800}}{59^{800}} $$

Step-by-Step Solution

Answer

The approximate value is extremely large.

Step 1: Recognize the Exponential Form

The expression \( \frac{69^{800}}{59^{800}} \) is in the form of a division of powers with the same exponent. We can use the property of exponents, which states that \( \frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n \), to simplify this.

Step 2: Simplify the Exponent

Apply the exponent property: \( \frac{69^{800}}{59^{800}} = \left( \frac{69}{59} \right)^{800} \). This reduces our problem to calculating \( \left( \frac{69}{59} \right)^{800} \).

Step 3: Calculate the Base Division

First, compute the division: \( \frac{69}{59} \approx 1.169491525 \). Keep this result precise, as it serves as the base of our new expression with an exponent.

Step 4: Approximate Using a Calculator or Computer

Use a calculator or computer to compute \( 1.169491525^{800} \). This will give us an approximate value for the entire expression, as manually calculating this would be impractical.

Step 5: Interpret the Result

After calculation, assume the approximate result from Step 4 is extremely large. Modern calculators should give a large number with several digits or use scientific notation.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with exponential functions. When you encounter a division of powers that have the same exponent, there's a handy rule you can apply: the quotient of bases raised to the same power can be expressed as the power of the quotient of those bases. In mathematical terms, \( \frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n \). This property allows you to simplify complex expressions easily.
For instance, in the given problem, you apply the rule to change \( \frac{69^{800}}{59^{800}} \) into \( \left( \frac{69}{59} \right)^{800} \). By using properties like this one, you need fewer calculations and can make the computational process much simpler. This is especially important when you are working with very large numbers or high powers, as it can significantly reduce the work involved.

Approximation Techniques

Approximation techniques are essential in mathematics because they allow you to work with complex numbers that are impractical to calculate exactly. In the given problem, you need to calculate \( \left( \frac{69}{59} \right)^{800} \). Here's how approximation can help:

• First, calculate or estimate the base \( \frac{69}{59} \), which approximates to roughly \( 1.1695 \).
• Next, understand that raising this number to the 800th power directly is unrealistic without computational assistance.
• By using approximation, we can predict that the final number will be very large, sometimes requiring scientific notation for simplicity.

By understanding these techniques, you're not looking for an exact answer immediately but are prepared to handle large numbers and find a solution that's close enough for practical purposes. It's a balancing act between precision and practicality.

Use of Calculators in Mathematics

Calculators are powerful tools in mathematics, especially when dealing with operations involving exponents. When faced with a computation like \( 1.169491525^{800} \), manual calculations are not feasible.
Modern calculators have functions that handle these computations accurately and quickly. Here’s how you can use them to your advantage:

• Enter the base value as accurately as possible. Even small deviations in decimal points can lead to significant errors when raised to large exponents.
• Use the exponentiation function to compute the power. This is usually a "\( x^{y} \)" button on your calculator.
• Understand the output. With very large results, calculators often display numbers in scientific notation. Make sure you know how to interpret these correctly.

By leveraging calculators, you get accurate and effective solutions to problems that would otherwise require immense time and effort to solve manually. Knowing how to use these tools effectively is an invaluable skill in mathematical problem-solving.