Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$32^{0.2}$$

Step-by-Step Solution

Answer

Using a calculator, \(32^{0.2} \approx 2.2973967\).

Step 1: Identify the Base and Exponent

In the expression \(32^{0.2}\), the base is 32, and the exponent is 0.2. This means we are finding the 0.2th power of 32, which can also be interpreted as the fifth root of 32 since 0.2 is equivalent to \(\frac{1}{5}\).

Step 2: Use a Calculator to Compute the Root

Enter the base, 32, into your calculator. Use the exponentiation or power function, and enter 0.2 as the exponent. This will compute \(32^{0.2}\).

Step 3: Read the Display

After performing the calculation, read the number that appears on the calculator's display. It should give the value of \(32^{0.2}\) to the precision available to your calculator.

Step 4: Record the Result

Write down the complete number as shown on your calculator’s display. This is your answer for \(32^{0.2}\).

Key Concepts

Fifth Root

The concept of the fifth root is all about finding a number which, when multiplied by itself five times, results in the original number. In this exact scenario, finding the fifth root is equivalent to raising a number to the power of \( \frac{1}{5} \). Here's how it works: if we have a number like 32, the fifth root is a number that satisfies \( x^5 = 32 \).

• This process helps in simplifying numbers or expressing them in a different form.
• It’s especially useful in algebra, physics, and other scientific calculations.

When dealing with fractional exponents such as \( 32^{0.2} \), understanding that 0.2 is equivalent to \( \frac{1}{5} \) helps in translating the problem into finding a fifth root. This mathematical relationship transforms complex exponential problems into simpler root calculations which can be solved by calculators.

Calculator Usage

Using a calculator effectively can simplify complex mathematical expressions. Knowing how to properly input numbers and functions is essential, especially when working with exponents and roots. Here's a simple guide for using a calculator to find roots or powers, such as computing \( 32^{0.2} \):

• Start by entering the base number, which in this case is 32.
• Look for the exponentiation function, often denoted as '^[" button or function.
• Enter the exponent, 0.2, carefully. This represents the fractional power or root you need to compute.
• Execute the function to get the result. Ensure the calculator's precision setting is high enough to provide a detailed output.

With practice, you'll find that calculators are an indispensable tool for checking your work and ensuring accuracy in mathematical exercises.

Base and Exponent Identification

Understanding the components of an exponential expression is crucial for solving or evaluating it. An expression like \( 32^{0.2} \) consists of two main parts: the base and the exponent. Here's how to identify them:

• The base is the number that is being multiplied by itself, which is 32 in this case.
• The exponent, 0.2, indicates how many times the base is used in multiplication. Here, it represents the fifth root.

By clearly identifying the base and the exponent, you can better understand the expression and decide on the correct mathematical operation, such as using the exponentiation function on your calculator. This clarity is important for effectively solving problems and understanding outcomes.

Exponentiation Function

The exponentiation function is fundamental in mathematics for raising numbers to a particular power. To perform exponentiation, especially on calculators, understanding this function is crucial. Here is what you typically need to know:

• The function allows you to compute numbers such as \( 32^{0.2} \) by using the base number (32) and applying the exponent (0.2).
• It is often represented as 'power' or sometimes as a '^' symbol on calculators.
• Using the exponentiation function accurately requires attention to detail - correct input of the base and exponent is vital for obtaining the right result.

This function significantly simplifies the process of exponentiation, providing quick and accurate results that would otherwise be difficult to compute manually. It’s a powerful feature for tackling a variety of math problems efficiently.