Problem 51 - Algebra and Trigonometry

Question

Powers of Factorials Which is larger, \((100 !)^{101}\) or \((101 !)^{100} ?[\text { Hint: Try factoring the expressions. Do they }\) have any common factors?]

Step-by-Step Solution

Answer

\((101!)^{100}\) is larger than \((100!)^{101}\).

Step 1: Understand the Problem

We need to compare two large numbers: \((100!)^{101}\) and \((101!)^{100}\). Our goal is to determine which expression is larger.

Step 2: Simplify the Expressions

Write each expression in terms of factorials:- \((100!)^{101} = 100! \times 100! \times \ldots \times 100!\) (101 times)- \((101!)^{100} = 101! \times 101! \times \ldots \times 101!\) (100 times)We know that \(101! = 101 \times 100!\). We can rewrite \((101!)^{100}\) using this fact.

Step 3: Factor and Expand

Express \((101!)^{100}\) using \(101! = 101 \times 100!\):- \((101!)^{100} = (101 \times 100!)^{100} = 101^{100} \times (100!)^{100}\) Now, notice that \((100!)^{101} = 100! \times (100!)^{100}\).

Step 4: Compare the Expressions

Observe that both expressions have \((100!)^{100}\) as a common factor:- \((100!)^{101} = 100! \times (100!)^{100}\)- \((101!)^{100} = 101^{100} \times (100!)^{100}\)Thus, the problem reduces to comparing \(100!\) with \(101^{100}\).

Step 5: Make the Final Comparison

\(101^{100}\) is a large number as it involves multiplying 101 with itself 100 times, which is considerably larger than simply \(100!\), a factorial of much smaller numbers. Therefore, \(101^{100} \times (100!)^{100} > 100! \times (100!)^{100}\).

Step 6: Conclusion

Thus, \((101!)^{100}\) is larger than \((100!)^{101}\) by comparing the factors beyond the shared \((100!)^{100}\) term.

Key Concepts

Powers of Factorials

In mathematics, factorials are functions applied to numbers to find the product of an integer and all the integers below it. A power of a factorial is when this product is raised to a certain exponent. Consider \((100!)^{101}\) and \((101!)^{100}\). These represent a factorial of a number raised to another power:

• \((100!)^{101}\) means 100 factorial is multiplied by itself 101 times.
• \((101!)^{100}\) implies 101 factorial is multiplied by itself 100 times.

Understanding powers of factorials involves knowing both how factorials work and how powers expand these large numbers drastically. Factorials grow very fast, and raising them to a large power amplifies this effect, making expressions involving powers of factorials extremely large.

Comparison of Expressions

When comparing expressions like \((100!)^{101}\) and \((101!)^{100}\), it's crucial to break them down into simpler components. By expressing each factorial in terms of known mathematics, like identifying common factors, you can make the comparison more straightforward.For example, knowing that \(101! = 101 \times 100!\) helps simplify and factorize the second expression. Both expressions share a common component, \((100!)^{100}\), which can be factored out to allow simpler comparison of the remaining parts. This step is crucial in understanding which expression is larger without needing to calculate the entire factorial and power directly, which is computationally impractical.

Factorization

Factorization is the process of breaking down numbers or expressions into their simplest components or factors. In the context of factorials and powers, this involves expressing a seemingly complex expression in terms of its basic multiplicative components. With \((101!)^{100} = 101^{100} \times (100!)^{100}\), factorization reveals that the complexity of the expression is somewhat manageable, as it isolates the effect of the factor \(101^{100}\).Factorization simplifies comparisons and calculations by isolating variables and constants. This highlights which part of an expression is influential in increasing or decreasing the entire term's value, which often is not immediately obvious in its expanded form.

Large Numbers Comparison

Comparing large numbers like \(101^{100}\) and \(100!\) requires more than just arithmetic; it demands logical thinking and mathematical properties.

• \(101^{100}\) is exponential, meaning it's 101 multiplied by itself 100 times, resulting in an astronomically large number.
• \(100!\) represents the multiplicative series from 1 to 100, which while large, grows slower than an exponential growth dictated by powers.

In conclusion, when facing huge numbers, analyzing the type of mathematical growth they represent—factorial versus exponential—gives insight into which is larger. Exponentials often outpace factorials for comparisons involving similar-sized bases and exponents, a crucial insight for tackling similar mathematical challenges.