Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$.\left(\frac{1}{3}\right)^{\sqrt{6}}$$

Step-by-Step Solution

Answer

The approximate value is 0.10364.

Step 1: Express the Problem Form

We need to approximate the expression \(rac{1}{3}\) raised to the power of \(\sqrt{6}\). This can be written as \(\left(\frac{1}{3}\right)^{\sqrt{6}}\).

Step 2: Calculate \(\sqrt{6}\)

Use your calculator to find the square root of 6. Input '6' and press the square root (\(\sqrt{}\)) function to get an approximate value, such as 2.44949.

Step 3: Compute the Exponential Expression

Enter \(\frac{1}{3}\) into your calculator, raise it to the power of the approximate value of \(\sqrt{6}\) from Step 2. This involves using the power or exponent function usually depicted as '^'. Calculate \(\left(\frac{1}{3}\right)^{2.44949}\).

Step 4: Record Approximate Result

Upon performing the calculation in Step 3, round your result according to your calculator's display limit, for example, 0.10364.

Key Concepts

Calculator Approximation

Calculators are essential tools for finding approximations of complex numerical expressions. Often in mathematics, we encounter numbers that are not simple to compute like fractions raised to irrational exponents. Calculators help simplify this task, especially when dealing with power functions that are not integers or whole numbers.
Using a calculator for approximation involves a few easy steps:

• Identify the mathematical operation you need to perform, like square roots or exponentiations.
• Input the numbers accurately and use the proper function keys. For square roots, use the \(\sqrt{}\) button, and for powers, look for the '^' or similar power key.
• Ensure your calculator is set to display as many decimal places as possible for accuracy.

This approach delivers a highly precise numerical answer close to the exact mathematical result, which is often critical for advanced calculations.

Exponentiation

Exponentiation is the mathematical operation involving the raising of a base number to a certain power or exponent. This key mathematical operation is akin to repeated multiplication. For instance, raising \((\frac{1}{3})^{\sqrt{6}}\) involves multiplying \(\frac{1}{3}\) by itself \(\sqrt{6}\) times. This can be cumbersome to calculate manually, especially with non-integer powers.
To simplify:

• The base, \(\frac{1}{3}\), is the number we multiply by itself.
• The exponent, \(\sqrt{6}\), determines how many times the base is multiplied by itself, though in a conceptual manner here due to its non-integer nature.

Calculators are equipped to handle such complex exponentiations using functions like '^'. They allow you to quickly compute expressions with fractional and irrational exponents by following the same basic procedure of inputting the base, pressing the power function key, and entering the exponent.

Square Root Calculation

Square roots are special mathematical operations that find a number which when multiplied by itself gives the original number. In this exercise, calculating \(\sqrt{6}\) is the first crucial step. The notation \(\sqrt{n}\) involves finding such a number for \(n\).
Using a calculator simplifies this process:

• Press the square root button, typically marked by \(\sqrt{}\).
• Enter the number (6 in this case) into the calculator.
• Read the displayed result, for example, 2.44949, as the approximate square root of 6.

Calculating square roots using a calculator is straightforward, yet understanding the implication of the operation helps in grasping how it fits into more advanced equations like exponentiation.