Problem 46 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$(2 \pi)^{4 / 3}$$

Step-by-Step Solution

Answer

The result is approximately 10.053096491487338.

Step 1: Understand the Expression

We need to find the power of \((2\pi)^{4/3}\). This means we take 2 multiplied by \(\pi\) and then raise the result to the power of \(\frac{4}{3}\).

Step 2: Calculate the Base

Calculate the value of the base, \(2\pi\). Using a calculator, \(\pi\) is approximately 3.141592653589793. Thus, \(2\pi = 2 \times 3.141592653589793 = 6.283185307179586\).

Step 3: Raise to the Power

With the base calculated as approximately 6.283185307179586, raise this number to the power of \(\frac{4}{3}\). Using the calculator again, input the base and the exponent to get the result: \((6.283185307179586)^{4/3}\).

Step 4: Report the Result

After performing the calculation on the calculator, the display shows the result as approximately 10.053096491487338. This contains as many digits as typically shown by a scientific calculator.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a number, known as the base, is raised to the power of another number, called the exponent. This process is often used to solve problems involving growth and decay, as it shows how quantities change over time. For example:

• If you want to calculate compound interest, you can use an exponential function to determine how your investment grows over time.
• Exponential functions are also used in population studies to predict how numbers increase or decrease.

In our exercise, \((2\pi)^{4/3}\), we have a base of \(2\pi\) raised to the exponent \(\frac{4}{3}\). This means that we multiply \(2\pi\) by itself a fractional number of times. Calculators handle these computations easily, giving you precise results quickly.When you encounter an expression with fractional exponents like \((4/3)\), it involves taking the cube root (since \(3\) is in the denominator), and then raising that result to the fourth power (due to \(4\) in the numerator). Understanding this concept helps you effectively use scientific calculators for exponential tasks.

Scientific Calculations

Scientific calculations involve using specialized calculators designed to handle complex equations, including powers, roots, and transcendental numbers like \(\pi\). These calculators can:

• Store constants like \(\pi\) and \(e\), reducing manual input errors.
• Help visualize the input numbers with the right precision, often up to 15 decimal places.
• Perform operations involving large or extremely small numbers efficiently.

In exercises like the one provided, a scientific calculator becomes invaluable as it streamlines the process of calculating the power of an expression like \((2\pi)^{4/3}\). Calculators simplify this by allowing direct input of the expression, showcasing the result with high accuracy on the display. By understanding how to use the calculator for such expressions, you minimize calculation errors, ensuring more accurate results which are critical in scientific and engineering applications.

Pi (π) in Mathematics

Pi \((\pi)\) is a mathematical constant representing the ratio of a circle's circumference to its diameter. Its approximate value is \(3.14159\), but it is an irrational number, meaning it has an infinite number of decimal places without repeating.\(\pi\) appears in various mathematical formulas, from geometry to trigonometry, and is instrumental in representing periodic phenomena in physics. For computational purposes,

• Scientific calculators often have a built-in \(\pi\) function to ensure precision.
• When working with calculations like in the given exercise, using the calculator's \(\pi\) function helps maintain accuracy.

Understanding how to use \(\pi\) in calculations is essential for tackling problems in mathematics and science involving circles or waves. By integrating \(\pi\) within expression like \(2\pi\), students can efficiently calculate powers and roots when combined with other numbers. This highlights the importance of \(\pi\) in both theoretical and practical applications.