Problem 11 - A Graphical Approach to College Algebra

Question

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\left(\frac{1}{2}\right)^{\sqrt{2}}$$

Step-by-Step Solution

Answer

The approximation is 0.7071067812.

Step 1: Understand the Expression

The given expression is \( \left(\frac{1}{2}\right)^{\sqrt{2}} \). We need to find the value of this expression using a calculator.

Step 2: Calculate the Square Root

First, calculate \( \sqrt{2} \) using your calculator. This typically gives approximately 1.414213562.

Step 3: Compute the Power

Next, compute \( \left(\frac{1}{2}\right)^{1.414213562} \) using your calculator. Enter 0.5 raised to the power of the square root of 2.

Step 4: Record the Result

The calculator will display the result with its maximum number of decimal places. Depending on the calculator, this could vary slightly. For most calculators, the result will be approximately 0.7071067812.

Key Concepts

Decimal Places

When using a calculator to find an approximation, the number of decimal places refers to how many digits are shown after the decimal point in the result. Calculators typically display up to a certain number of decimal places to balance precision and clarity. Understanding this can help you know how close your approximation is to the true value.

In the given exercise, we calculate \( \\left(\frac{1}{2}\right)^{\sqrt{2}} \). After computing, you'll notice the calculator will display the result to its maximum number of decimal places. This precise display helps ensure the result is accurate enough for practical use, though never absolutely exact.

• Make note of which calculator model you are using, as display capabilities can vary.
• Keep in mind the more decimal places shown, the more precise the approximation will be.
• The concept of decimal places is essential when dealing with scientific and technical calculations, where precision matters a lot.

By becoming familiar with your calculator's decimal limit, you can better interpret your results.

Power Calculation

Power calculation is a fundamental mathematical operation involving raising a number to the exponent of another number. In this exercise, we worked with the expression \( \\left(\frac{1}{2}\right)^{\sqrt{2}} \). To perform this operation, we are essentially multiplying the base number, 0.5, by itself \( \\sqrt{2} \)-times, which can be calculated as a decimal equivalent.

Using a calculator:

• Input the base number, 0.5.
• Select the power function, often represented by a "^" or "y^x" button.
• Enter the exponent, which here is \(\sqrt{2}\approx 1.414213562\).
• Compute to find the result.

This sequence helps you efficiently use your calculator, ensuring you correctly apply the power operation. Understanding power calculation is crucial because it shows how exponential growth or decay can strongly affect numbers, especially in areas like finance, physics, or computing.

Square Root

Finding a square root is a common mathematical operation where you identify a number that, when multiplied by itself, gives the original number. In this exercise, we needed to find \( \\sqrt{2} \), a number which is approximately 1.414213562. This value is irrational, meaning it cannot be represented exactly as a simple fraction, and its decimal representation is non-repeating and infinite.

To find a square root on a calculator:

• Locate the square root function, usually symbolized by \( \\sqrt{} \) or a button labeled as such.
• Enter the number, 2, that you want to find the square root of.
• Press the square root button to calculate.
• The calculator will return an approximation of about 1.414213562.

Understanding square roots is critical as they appear frequently in geometry, algebra, and various practical applications, such as determining dimensions or solving quadratic equations.