Problem 56 - Precalculus Mathematics for Calculus

Question

Find the product of the numbers $$ 10^{1 / 100}, 10^{2 / 10}, 10^{3 / 10}, 10^{4 / 10}, \ldots, 10^{19 / 10} $$

Step-by-Step Solution

Answer

The product is \(10^{18.91}\).

Step 1: Understand the Problem

We are asked to find the product of numbers in the series: \(10^{1/100}, 10^{2/10}, 10^{3/10}, \ldots , 10^{19/10}\). Each term can be represented in the form \(10^{n/10}\) except the first one, which is \(10^{1/100}\). To better analyze it, convert the exponents into a more manageable form.

Step 2: Rewrite Exponents

Rewrite the series with clear exponents: \(10^{1/100}, 10^{20/100}, 10^{30/100}, \ldots, 10^{190/100}\). This simplifies the terms to \(10^{1/100}, 10^{1/5}, 10^{3/10}, \ldots, 10^{19/10}\).

Step 3: Use Properties of Exponents

When multiplying numbers with the same base, add the exponents: \((10^a \times 10^b \times \cdots) = 10^{a+b+\cdots}\). We need to find the sum of the exponents of all terms.

Step 4: Find the Sum of Exponents

Sum the exponents: \(\frac{1}{100} + \frac{20}{100} + \frac{30}{100} + \cdots + \frac{190}{100}\). Factoring out \(\frac{1}{100}\), the sum becomes \(\frac{1}{100}(1 + 20 + 30 + \cdots + 190)\).

Step 5: Compute the Series Sum

The sequence \(1, 20, 30, \ldots, 190\) is not an arithmetic series. Recognize an error: the correct sequence is \(\frac{1}{100} + \frac{2}{10} + \frac{3}{10} + \cdots + \frac{19}{10}\). This matches the general term \(10^{n/10}\) for \(n = 1, 2, \ldots, 19\).

Step 6: Calculate the Total Exponent Sum

Sum of exponents: \(\frac{1}{100} + (\frac{2}{10} + \frac{3}{10} + \cdots + \frac{19}{10})\). Simplifying further: \(\frac{1}{100} + \frac{2+3+\cdots+19}{10}\). Calculate the sum of integers from 2 to 19 using the formula \(\frac{n(n+1)}{2} - 1\) for \(n=19\).

Step 7: Solve the Summation

Calculate the integer series: \(\frac{19(20)}{2} - 1 = 189\). Exponent sum becomes: \(\frac{1}{100} + \frac{189}{10}\).

Step 8: Simplify Final Exponent

The exponent simplifies to \(\frac{1}{100} + 18.9 = 18.91\).

Step 9: Find the Product

The product of the given series is \(10^{18.91}\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions are expressed in the form \(a^x\), where \(a\) is the base and \(x\) is the exponent. Exponential functions are widely applicable in fields like finance for compound interest calculations and in sciences for modeling growth or decay processes.

• They grow rapidly as the exponent increases.
• The domain of exponential functions is all real numbers.
• The range is positive real numbers for \(a > 1\).

In our original exercise, the exponential function is expressed in base 10 with various fractional exponents, such as \(10^{1/100}\) and \(10^{19/10}\). Understanding how these exponents work helps in calculating products and powers efficiently.

Properties of Exponents

The properties of exponents are essential rules that help in simplifying mathematic operations involving exponents. The main properties include:

• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Power of a power: \((a^m)^n = a^{m \times n}\)
• Power of a product: \((ab)^n = a^n \times b^n\)
• Zero exponent: \(a^0 = 1\)

In our problem, the product of exponential terms \(10^{n/10}\) involves using the product of powers property. By adding all the exponents together, you easily determine that the overall exponent for the product is \(18.91\). Knowing these properties makes it easier to handle more complex problems involving exponential functions.

Series and Sequences

A series is the sum of the terms of a sequence. Sequences are ordered lists of numbers, which can follow a particular pattern or rule. Series can either be finite or infinite.In the given problem, we have a finite sequence with exponential terms. We corrected the sequence as \(\frac{1}{100}, \frac{2}{10}, \frac{3}{10}, \ldots, \frac{19}{10}\). These terms define a sequence where each exponent builds on the previous one, based on the formula \(10^{n/10}\).
To find the sum \(1 + 2 + \ldots + 19\), we used the formula for an arithmetic series. However, our adjustment focused on only summing the terms \(2\) to \(19\), leading us to calculate \(\frac{18.9}{10}\), simplifying our exponent operations to reach \(18.91\). Understanding sequences and series allows you to handle the sum of complex patterns efficiently.