Problem 56 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The product is \(10^{19}\).

Step 1: Understanding the Problem

We need to find the product of a sequence of exponentiated numbers: \( 10^{1/10}, 10^{2/10}, \ldots, 10^{19/10} \).

Step 2: Write the Product Expression

The product of the given terms is written as: \(10^{1/10} \times 10^{2/10} \times 10^{3/10} \times \ldots \times 10^{19/10}\).

Step 3: Simplify Using Properties of Exponents

Using the property of exponents that allows us to add exponents when multiplying like bases, \(a^{m} \times a^{n} = a^{m+n}\), we can write this as \(10^{(1/10 + 2/10 + 3/10 + \ldots + 19/10)}\).

Step 4: Calculate the Sum of the Exponents

The exponents form an arithmetic series: \(1/10, 2/10, \ldots, 19/10\). The sum of this series is \(\sum_{k=1}^{19}\frac{k}{10} = \frac{1}{10}(1+2+3+\ldots+19)\).

Step 5: Calculate the Arithmetic Series

Use the formula for the sum of the first \(n\) natural numbers: \(\sum_{k=1}^{n}k = \frac{n(n+1)}{2}\). For \(n=19\), this is \(\frac{19 \times 20}{2} = 190\).

Step 6: Complete the Simplification

Substitute the sum into the expression: \(\frac{190}{10} = 19\). Thus, the product simplifies to \(10^{19}\).

Step 7: Final Conclusion

The product of the sequence \(10^{1/10}, 10^{2/10}, \ldots, 10^{19/10}\) is \(10^{19}\).

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous one. Think of it as a linearly growing series of numbers. For example, with a difference of 1, you would see terms like 1, 2, 3, and so on.

• In the context of our problem, the exponents form the arithmetic series \( \frac{1}{10}, \frac{2}{10}, \ldots, \frac{19}{10} \). These are increasing by \( \frac{1}{10} \) each step.
• This regular structure helps to simplify how we sum the terms, critical for simplifying expressions involving these terms.

Knowing how to handle an arithmetic series is essential for managing operations where these series appear, such as finding sums or products involving these sequences.

Properties of Exponents

Exponents have several key properties that make them easier to work with. These rules help transform potentially complex expressions into more manageable forms.

• Product of Powers Property: When multiplying two powers with the same base, you simply add the exponents. For example: \( a^m \times a^n = a^{m+n} \).
• This simplifies exponential expressions significantly when you're dealing with sequences like the one in our problem as \(10^{1/10} \times 10^{2/10} \times \ldots = 10^{\text{sum of exponents}}\).

Understanding these properties equips you with tools to tackle various problems involving exponents without intimidation.

Sum of Natural Numbers Formula

The sum of natural numbers is a crucial formula that provides the total of all consecutive numbers up to a specific one:

• Formula: \( \sum_{k=1}^n k = \frac{n(n+1)}{2} \), where \( n \) is the last number in the series.
• In our exercise, we used this formula to find the sum of numbers from 1 to 19. This is because adding these numbers directly gives us the sum needed for our arithmetic series \(1 + 2 + 3 + \ldots + 19\).
• Once summed, this value helps simplify the exponentiation to \( \frac{190}{10} \) leading to a final expression of \( 10^{19} \).

Grasping this formula is valuable as it extends to many mathematical scenarios involving natural numbers and can save time and effort in numerous mathematical calculations.