Problem 33 - 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. $$\sqrt[3]{-125}$$

Step-by-Step Solution

Answer

The cube root of -125 is -5.

Step 1: Identifying the Problem

The given problem is to find the cube root of -125. This is represented mathematically as \( \sqrt[3]{-125} \).

Step 2: Understanding Cube Roots

A cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\). In this problem, we need to find a number whose cube is -125.

Step 3: Using a Calculator

Turn on your calculator, and use the cube root function. Depending on your calculator model, you might find this by pressing a 'root,' 'shift,' or 'function' key followed by the number '3' and then inputting -125.

Step 4: Calculating the Cube Root

After inputting the function \( \sqrt[3]{-125} \) into the calculator, press the equals button to get the result.

Step 5: Reading the Result

The calculator will display the result, which should be -5, because \((-5) \times (-5) \times (-5) = -125\).

Key Concepts

Calculator Usage for Cube Roots

Using a calculator can make finding cube roots super simple! When tasked with finding the cube root of a number, like \(-125\), a calculator can do the heavy lifting. First, you need to locate the cube root function on your calculator.

• On most calculators, it's accessible via a key labeled 'root', 'shift', or 'function'.
• Pressing this will usually involve entering '3', as it's the cube root you're after.

After setting up the calculator to find the cube root, enter the number \(-125\). Finally, hit the equals button. Voila! The result should appear instantly on your screen. Calculators are accurate and efficient, plus they help in providing the exact number of digits as displayed.

Understanding Negative Numbers

Negative numbers can initially seem confusing, especially when combined with roots. However, understanding them can significantly ease your calculations. Let's clarify things a bit.When dealing with cube roots, remember:

• Negative numbers have real cube roots, since multiplying three of the same negative numbers results in another negative number.

For example, \((-5) \times (-5) \times (-5) = -125\), which shows that the cube root of \(-125\) is \(-5\). Unlike square roots, which aren't real for negative numbers, cube roots work smoothly with them. It’s all about balance between the negative signs, turning a complex-looking number into a simple concept.

Mastering Exponents with Cube Roots

Exponents play a vital role in understanding roots, especially cube roots. It's like finding the opposite of a power. When you need the cube root of a number, didn't it first come from being multiplied by itself twice more?

• The cube root of a number \(x\), expressed as \(\sqrt[3]{x}\), solves \(y^3 = x\) for \(y\).

Consider \(x = -125\). Here, finding \(\sqrt[3]{-125}\) finds the value of \(y\) that satisfies \(y^3 = -125\). This is a crucial concept, as recognizing \((-5)^3 = -125\) not only solves the problem but also aids in grasping how power and roots work together in harmony.