Problem 41 - Algebra 2

Question

Use a calculator to approximate each value to three decimal places. $$ \sqrt[3]{59} $$

Step-by-Step Solution

Answer

The cube root of 59 is approximately 3.884.

Step 1: Understand the Expression

The expression \( \sqrt[3]{59} \) represents the cube root of 59. Our goal is to find a number that, when multiplied by itself three times, gives approximately 59.

Step 2: Use a Calculator for Approximation

To approximate \( \sqrt[3]{59} \), input the expression into a scientific calculator. If using a standard calculator, you can access the cube root function, often labeled as \( \text{cbrt} \) or something similar. Input 59 and press the cube root key.

Step 3: Record the Result

After performing the calculation, the cube root of 59 is approximately 3.884. Ensure that the result is rounded to three decimal places for accuracy.

Key Concepts

Approximation

When working with cube roots, especially those that do not result in whole numbers, we often rely on approximation. Approximating a number means finding a value that is close enough to the exact answer for practical purposes. In this exercise, our goal is to approximate \( \sqrt[3]{59} \) to three decimal places.

Approximations are particularly useful when the exact value is irrational, meaning it cannot be expressed as an exact fraction, like the cube root of 59. Thus, we use decimal form to represent these numbers as closely as possible.

The precision of our approximation depends on how many decimal places we use. Here, the task is to give the result to three decimal places for clarity and usability. This precision offers a balance between simplicity and accuracy.

Scientific Calculator

Scientific calculators are powerful tools when dealing with complex mathematical operations, such as finding cube roots. They extend the functionality of standard calculators by including capabilities beyond basic arithmetic. These include functions like exponentiation, logarithms, and roots.

For the task of approximating \( \sqrt[3]{59} \), a scientific calculator is invaluable. It typically features a specific button or function for calculating cube roots. This is commonly denoted by 'cbrt' or sometimes appears as a shifted function on a regular root button.

To find the cube root using a scientific calculator, you simply input the number (in this case, 59) and use the cube root function. This method is quick and reduces the likelihood of human error that might occur if attempting a manual calculation.

Cube Root Function

The cube root function is a mathematical function used to determine a number that, when multiplied by itself three times, equals a given value. Mathematically, for a number \( x \), the cube root is represented as \( \sqrt[3]{x} \).

Understanding this function is crucial when solving problems involving volumes and other practical applications where three-dimensional measurements are considered. For example, if you know the volume of a cube, finding its side length involves computing the cube root of that volume.

The process of finding a cube root is not as intuitive as squaring a number, which makes the cube root function significant in computing. With the help of tools like scientific calculators, finding the cube root of numbers like 59 becomes straightforward, giving us a result such as approximately 3.884.