Problem 9 - 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. $$2^{\sqrt{10}}$$

Step-by-Step Solution

Answer

2 raised to the power of the square root of 10 is approximately 9.050084.

Step 1: Identify the expression to evaluate

The expression given is \( 2^{\sqrt{10}} \). We need to compute the value of this exponential expression using a calculator.

Step 2: Enter the expression in the calculator

On your calculator, locate the power or exponent function. Enter the base value which is 2, and then enter the exponent, which is \( \sqrt{10} \). Make sure you perform the square root calculation first if your calculator does not support direct inputs.

Step 3: Calculate \( \sqrt{10} \)

Calculate \( \sqrt{10} \) using your calculator. The approximated value should be around 3.162277660.

Step 4: Input the exponential calculation

With \( \sqrt{10} \) calculated, input \( 2^{3.162277660} \) into the calculator to find \( 2^{\sqrt{10}} \).

Step 5: Obtain the final result

Compute the value of \( 2^{3.162277660} \) using your calculator. The result should display the maximum number of decimal places, potentially resulting in approximately 9.050084.

Key Concepts

Calculator Usage

Using a calculator efficiently for mathematically involved expressions like exponential ones is crucial, especially in exams or homework tasks. It's not just about pressing buttons; it's about understanding where each function fits into the calculation process. To solve problems like finding the value of \( 2^{\sqrt{10}} \), you start by identifying the need for functions like square root and exponentiation.

Here’s a straightforward approach:

• Find and use the square root function first if your calculator requires separate steps for such operations.
• Input the base value and exponent into your calculator using the exponentiation function. Most calculators have a specialized button, like \( ^{\wedge} \) or \( \text{exp} \), for this purpose.

Many scientific calculators can handle nested calculations, which means you can input the base and the square root of the exponent directly, streamlining the process. Understanding how your specific calculator works can greatly improve accuracy and speed in calculations.

Square Roots

Square roots are a fundamental concept in mathematics, representing one of two equal factors of a number. In this problem, you are tasked with finding the square root of 10 in order to use it as an exponent. When you take the square root of 10, denoted as \( \sqrt{10} \), you are seeking a number that, when multiplied by itself, equals 10.

On most calculators:

• Locate the square root button, often represented by the symbol \( \sqrt{} \) or written as \( \text{sqrt} \).
• Enter 10 and press the square root function button to get an immediate result.
• The calculator should display approximately 3.162277660, the decimal approximation of \( \sqrt{10} \).

By understanding the process and purpose of calculating square roots, you can better approach complex math problems. It's not just about the calculation but interpreting what the result tells you, particularly when it's used as a power in further computations.

Decimal Approximation

Decimal approximation is crucial when dealing with irrational numbers, as in calculating \( 2^{\sqrt{10}} \). The expression \( \sqrt{10} \) itself is an irrational number, meaning it cannot be represented exactly as a fraction and its decimal form is non-repeating and non-terminating. Thus, we use decimal approximations for practical calculations.

Why decimal approximation?

• It provides a manageable way to represent irrational numbers.
• Calculators display results in decimal form to the highest precision possible.

In practice, after calculating \( \sqrt{10} \) as approximately 3.162277660, enter this decimal as the exponent to base 2. The result, found using your calculator, is about 9.050084 with the maximum precision your calculator allows. Though this is an approximation, it is sufficiently precise for most practical needs.