Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Find a decimal approximation of each root or power Round answers to the nearest thousandth. $$\sqrt[3]{91}$$

Step-by-Step Solution

Answer

The cube root of 91 is approximately 4.498.

Step 1: Understanding the Problem

We need to find a decimal approximation for the cube root of 91, rounded to the nearest thousandth.

Step 2: Estimating the Cube Root

The cube root of 91 (\( \sqrt[3]{91} \)) falls between two integers. Since \(4^3 = 64\) and \(5^3 = 125\), it is clear that \(\sqrt[3]{91}\) is between 4 and 5.

Step 3: Using a Calculator

Utilize a calculator to compute \(\sqrt[3]{91}\) more accurately. Enter the function into the calculator and find the decimal result.

Step 4: Rounding the Result

The calculated cube root of 91 is approximately 4.49794. Round it to the nearest thousandth, giving a result of 4.498.

Key Concepts

Decimal Approximation

When dealing with roots or powers, it's often necessary to find a decimal approximation first. This is especially true for numbers like cube roots, which are not perfect cubes. Decimal approximation involves expressing an irrational number—which can't be exactly represented on paper due to its non-repeating, non-terminating nature—in a simplified numeric form that is close to the actual value. For example, when finding \[ \sqrt[3]{91} \]we know the exact decimal will continue indefinitely, but we approximate it to a manageable number of decimal places, making it easier to grasp and work with in practical terms.

Rounding

Rounding is a mathematical technique used to simplify numbers while retaining their approximate value. When asked to round a number to the nearest thousandth, you are asked to keep only three decimal places. To do this: - Look at the fourth decimal number. - If it's 5 or more, round the third decimal place up by one. - If it's less than 5, leave the third number as is. For 4.49794, the fourth decimal is nine, so we increase the third decimal place from seven to eight. The rounded result is 4.498.

Calculator

Using a calculator is a great way to handle expressions that include roots, like the cube root of 91. Calculators can compute complex equations quickly and accurately, providing precise decimal approximations. Here's how you can use one for finding cube roots:- Access the cube root function, often labeled as \( x^{1/3} \) or \( \sqrt[3]{x} \).- Enter the number 91.- Execute the operation to display the cube root result.This method gives you 4.49794as the decimal approximation of \( \sqrt[3]{91} \). Use this accurate number for further analysis or rounding as needed.

Estimation

Before using a calculator to find an exact decimal approximation, estimation can help you set a range for the possible value. This is useful for cube roots where the exact result might not be readily apparent. Consider the number 91:- Recognize that \(4^3 = 64\) and \(5^3 = 125\), so you know \(\sqrt[3]{91}\) is between 4 and 5.By narrowing down the range, you gain a perspective on what the value should be close to, aiding in sanity-checking any calculations and ensuring they are within a realistic scope. This step emphasizes the importance of having a rough idea before diving into precise calculations.