Finding the Cubed Root of -64: A Step-by-Step Guide

Have you ever wondered how to find the cubed root of a negative number, like -64? It may seem tricky at first, but with a step-by-step guide, you'll be able to calculate it in no time! So, let's dive into the world of imaginary numbers and explore the fascinating concept of finding the cubed root of -64.

Step 1: Understanding Imaginary Numbers

Before we begin, it's important to understand the concept of imaginary numbers. Imaginary numbers are numbers that, when squared, result in a negative value. They are represented by the symbol 'i', where i = √(-1). In our case, we'll be dealing with the imaginary number i.

Step 2: Breaking Down the Problem

To find the cubed root of -64, we need to first break down the problem into simpler steps. Let's represent the cubed root of -64 as 'x'. So, we can write the equation as:

x³ = -64

Step 3: Introducing the Cube Root of -1

Here's where things get interesting! The cube root of -1 is a special imaginary number that is equal to (-1)^(1/3). It can be expressed as:

∛(-1) = -1

Step 4: Rewriting the Equation

Now, we can rewrite our original equation using the cube root of -1:

x³ = -64

(∛(-1))³ * x³ = -64

Step 5: Simplifying the Equation

Simplifying the equation, we get:

(-1) * x³ = -64

Step 6: Isolating x

To isolate x, we need to divide both sides of the equation by -1:

(-1) * x³ / (-1) = -64 / (-1)

x³ = 64

Step 7: Finding the Cube Root

Finally, we can find the cube root of 64 by taking the cube root of both sides of the equation:

∛(x³) = ∛64

x = 4

Conclusion

So, the cubed root of -64 is 4. Remember, when dealing with imaginary numbers, it's all about understanding the concept and following the steps carefully. With practice, you'll be able to conquer any cubed root problem that comes your way!