Problem 73 - Calculus with Concepts in Calculus

Question

Modify the expression, and then find its approximate value by calculator or computer. $$ \frac{(0.123)^{9000}}{(0.125)^{9000}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(0.984^{9000}\), which is approximately 0.

Step 1: Rewrite the Problem Statement

The original expression is \( \frac{(0.123)^{9000}}{(0.125)^{9000}} \). Since the exponent is the same for both the numerator and the denominator, we can simplify this expression.

Step 2: Simplify the Expression

We can simplify \( \frac{(0.123)^{9000}}{(0.125)^{9000}} \) to \( \left(\frac{0.123}{0.125}\right)^{9000} \). This uses the property \( \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n \).

Step 3: Calculate the Base Fraction

Compute the base of the simplified expression: \( \frac{0.123}{0.125} \). This gives us \( \frac{0.123}{0.125} = 0.984 \).

Step 4: Evaluate the Expression Using a Calculator

Now compute \( 0.984^{9000} \). Using a calculator or computer, you will find that \( 0.984^{9000} \) is approximately \( 0 \). This is because \( 0.984 \) raised to such a large power as 9000 becomes a very small number close to zero.

Key Concepts

Simplifying Expressions

Sometimes, math expressions can look complex at first glance. Simplifying them can make them more manageable and easier to understand. In cases like \( \frac{(0.123)^{9000}}{(0.125)^{9000}} \), you can use rules of exponents to simplify the problem by dealing with the numbers in an easier way.

• When the bases of the numbers in the numerator and denominator are raised to the same power, you can simplify this using the rule \( \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n \).
• This converts both the numbers into a single expression where the power applies to the result of their division. In our exercise, this translates to simplifying the task into finding \( \left(\frac{0.123}{0.125}\right)^{9000} \).

Breaking down the expression in this manner reduces the need for unnecessarily calculating large powers of cumbersome decimals separately. Instead, you just need to divide and then apply the exponent, easing the process considerably.

Properties of Exponents

Understanding the properties of exponents is essential for simplifying expressions in mathematics. One such property observed in our example is the ability to manipulate expressions where an exponent is applied to both the numerator and the denominator.

• The property \( \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n \) is particularly useful when solving problems like these because it allows you to compare the base values before exponentially expanding them.
• Essentially, this rule helps to condense the operation from a potentially complex equation into a much simpler base calculation followed by raising it to the required power.

By simplifying using this property, you not only make calculations easier but also better understand the relationship between similar powers.

Numerical Approximation

In mathematics, especially when working with very large or very small numbers, exact calculations aren't always feasible. This is where numerical approximation comes into play, providing a way to estimate results.

• For the expression \( 0.984^{9000} \), calculating it exactly is very complex without computational aid. However, we know that values slightly less than 1, when raised to very high powers, will approach zero.
• Hence, using a calculator or computer to find the approximate value shows \( 0.984^{9000} \) as essentially zero.

Understanding how to use and when to rely on numerical approximations is a crucial skill as it allows for quick, reasonable estimations in problem-solving, often necessary in more advanced mathematics or real-world applications.