Problem 34 - A Graphical Approach to College Algebra

Question

Use a calculator to find each root or power. Give as many digits as your display shows. $$\sqrt[5]{-243}$$

Step-by-Step Solution

Answer

The fifth root of -243 is approximately -3.0451 depending on the calculator's precision.

Step 1: Understanding the Problem

We need to find the fifth root of -243 using a calculator. This means we are looking for a number that multiplied by itself five times equals -243.

Step 2: Using the Calculator

On your calculator, enter the value -243 first. Then, look for the function that allows you to find the nth root, which is usually marked as a general root or `x√y` key. Enter 5 as the root value.

Step 3: Calculating the Fifth Root

Allow the calculator to process the inputs. The display will show a number, which is the fifth root of -243. This number has many decimal places depending on your calculator's display.

Key Concepts

Calculator Usage

Calculators are powerful tools for solving complex mathematical problems like finding nth roots. When tackling a problem like finding the fifth root of \(-243\), a calculator makes the process quicker and more accurate.
It's essential to understand how to input values for such calculations. Most modern calculators have a function for evaluating roots. This is often denoted by the symbol `\(x\sqrt{y}\)`, or might be available under a menu labeled as "math" or "functions."
To use a calculator for finding the fifth root:

• First, enter the number you are finding the root of, which in this case is \(-243\).
• Next, activate the root function, often using a specific button or sequence.
• Input the degree of the root, such as \(5\) for the fifth root.
• Press enter or equals to get the result.

Practicing these steps will make you comfortable with finding any root using a calculator, saving time and reducing errors.

Exponentiation

Exponentiation is a mathematical operation that involves multiplying a number by itself a certain number of times. When it comes to roots, it's essentially performing the inverse operation.
Taking the fifth root of a number like \(-243\) is asking: which number raised to the power of 5 equals \(-243\)?
In other words, if we find a number \(x\) such that:\[ x^5 = -243 \]Then, \(x\) is the fifth root of \(-243\).
Understanding this concept helps when dealing with different degrees of roots. Any number raised to a power is the process of repeated multiplication, and roots simply undo this operation.
Using this relationship can help in verifying answers provided by calculators. You can backtrack the calculator’s output by raising it to the fifth power to ensure the result equates to the original number.

Decimal Representation

When you calculate roots using a calculator, the result is often a long decimal number. Due to limited display, calculators show a precise, decimal approximation of the exact root.
For fifth roots, like \(\sqrt[5]{-243}\), the decimal display might extend several digits beyond the decimal point, providing a very accurate result.
As an example, the fifth root of \(-243\) is approximately \(-3.034\) because when multiplied by itself five times, it will equal \(-243\).
Understanding decimal representations is crucial for evaluating the precision of your answers. More digits mean a more precise answer, allowing for better estimations in further calculations or practical applications. If precision is vital for homework or projects, copy the entire decimal output given by your calculator.