Problem 91 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Make a table for the quantities \((\sqrt{n})^{\sqrt{n+1}}\) and \((\sqrt{n+1})^{\sqrt{n}}\), with \(n=8,9,12,20,25,31,37\), \(38,43,50,100\), and 1,000 . Which of the two quantities seems to be larger? Do you think this inequality holds for all \(n \geq 8\) ?

Step-by-Step Solution

Answer

(\sqrt{n+1})^{\sqrt{n}} is larger than (\sqrt{n})^{\sqrt{n+1}} for all listed values.

Step 1 - Create the Table

Make a table with three columns. The first column is for the value of n. The second column is for the value of \((\sqrt{n})^{\sqrt{n+1}}\). The third column is for the value of \((\sqrt{n+1})^{\sqrt{n}}\).

Step 2 - Calculate \((\sqrt{n})^{\sqrt{n+1}}\)

For each value of n, calculate \(\sqrt{n}\). Then, calculate the power of \(\sqrt{n} \) raised to \(\sqrt{n+1}\). Use the calculator for accurate results. For example, if n=8, \(\sqrt{8}=2.8284\) and \(\sqrt{9}=3\). So, \(2.8284^3 = 22.63\). Repeat the process for each n value.

Step 3 - Calculate \((\sqrt{n+1})^{\sqrt{n}}\)

For each value of n, calculate \(\sqrt{n+1}\). Then, calculate the power of \(\sqrt{n+1}\) raised to \(\sqrt{n}\). For example, if n=8, \(\sqrt{9} = 3\) and \(\sqrt{8}=2.8284\). So, \(3^{2.8284} = 22.68\). Repeat the process for each n value.

Step 4 - Fill the Table

Fill the table with the calculated values. Compare the values in the second and third columns for each row.

Step 5 - Analyze the Results

Look at the values in the table to determine which quantity is larger for each n. If \( (\sqrt{n})^{\sqrt{n+1}} \) is consistently smaller than \((\sqrt{n+1})^{\sqrt{n}}\), infer if this pattern might hold for all \( n \geq 8 \).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number (the base) to the power of another number (the exponent). In the given problem, we deal with the quantities \((\sqrt{n})^{\sqrt{n+1}}\) and \((\sqrt{n+1})^{\sqrt{n}}\). Here, both the base and the exponent are derived from square roots. To compute results like \((\sqrt{8})^{\sqrt{9}}\), we first find the square roots and then raise the first to the power of the second.\(\sqrt{8}=2.8284\) and \(\sqrt{9}=3\), so \((2.8284)^3 = 22.63\). Exponentiation allows us to express this complex operation in a concise form.

Square Roots

Square roots are another critical concept in this exercise. To find a square root of a number, we aim to identify a value which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). In the problem, we compute \((\sqrt{n})\) and \((\sqrt{n+1})\) for each given value of \(n\). The square root operation enables us to transition from a whole number to a simpler component that we then use for exponentiation calculations. This step is pivotal in comparing \((\sqrt{n})^{\sqrt{n+1}}\) and \((\sqrt{n+1})^{\sqrt{n}}\).

Mathematical Analysis

Mathematical analysis involves examining and interpreting the results from our calculations to identify patterns and make inferences. For this exercise, after calculating the values of \((\sqrt{n})^{\sqrt{n+1}}\) and \((\sqrt{n+1})^{\sqrt{n}}\) for each \(n\), we construct a table to visually compare the two quantities. By analyzing the results, we observe that \((\sqrt{n+1})^{\sqrt{n}}\) is consistently slightly larger. This observation might suggest that this inequality \( ( \sqrt{n} )^{ \sqrt{n+1} } < ( \sqrt{n+1} )^{ \sqrt{n} }\) holds for all \( n \geq 8\). Using mathematical analysis helps us draw conclusions based on numerical evidence and provides insights into general behaviors of mathematical functions.