Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Find a decimal approximation of each root or power. Round answers to the nearest thousandth. \(29^{1 / 3}\)

Step-by-Step Solution

Answer

The cube root of 29 is approximately 3.072.

Step 1: Understand the Problem

You need to find the cube root of 29. Mathematically, this is expressed as \(29^{1/3}\). The goal is to find this value and round it to the nearest thousandth.

Step 2: Use a Calculator

To find the decimal approximation of \(29^{1/3}\), use a calculator that has a function for computing roots or powers. Entering this into a scientific calculator will give you the decimal value.

Step 3: Calculate the Decimal Value

Enter \(29^{1/3}\) into your calculator. You should obtain approximately 3.072.

Step 4: Round to the Nearest Thousandth

The calculator output, 3.072, is already to the nearest thousandth since the thousandth place is 3 decimal places.

Key Concepts

Decimal Approximation

When working with roots and powers, such as finding the cube root of 29 (\(29^{1/3}\)), we often end up with a number that is not a simple fraction or whole number. This is where decimal approximation helps. Instead of expressing numbers exactly, we approximate them to a specific number of decimal places.
Let's consider what a decimal approximation entails:

• It involves representing a number in a form that is easier to understand, using decimals instead of fractions or irrational numbers.
• It is crucial when numbers cannot be expressed accurately because they don't have finite decimal expansions.

So, when you calculate \(29^{1/3}\) using a calculator, you might get a more extended decimal like 3.071850324. But for practical reasons, we need the approximation to be manageable, leading us to the concept of rounding.

Rounding

Rounding numbers is helpful for simplifying calculations and making them more digestible. When instructed to round to the nearest thousandth, it means trimming or adding to your decimal so it ends at three digits past the decimal point.
Here's how you can achieve this effectively:

• Identify the number to the right of your rounding digit (the thousandth place).
• If this number is 5 or greater, round up by adding one to the thousandth digit.
• If it is less than 5, round down or keep the thousandth digit as it is.

In our exercise, the cube root of 29 gives approximately 3.071850324. To round to the nearest thousandth, consider the fourth decimal place (8), and since it's 5 or greater, 3.072 is your rounded result.

Calculator Usage

A calculator becomes a very powerful tool when dealing with non-standard roots and powers which cannot be easily calculated mentally or on paper. Using a calculator for finding cube roots or an exponent like \(29^{1/3}\) is very simple with these steps:

• First, ensure your calculator is set to the correct mode for applying roots or calculating powers.
• Input the base number, 29 in this case, into the calculator.
• Use the cube root or power function, which might be represented as \(\sqrt[3]{}\) or \(\wedge(1/3)\).
• Execute the function to receive a decimal approximation of the cube root.

For example, entering \(29^{1/3}\) in a scientific or a graphing calculator will yield approximately 3.072. By understanding how to operate your calculator effectively, these tasks become quick and easy, saving time and reducing errors.