What Is The Following Product?$\sqrt[3]{24} \cdot \sqrt[3]{45}$A. $\sqrt[3]{69}$ B. $4(\sqrt[3]{6}$\] C. $6(\sqrt[3]{5}$\] D. $6(\sqrt[3]{10}$\]

**What is the Following Product? A Step-by-Step Guide to Solving the Equation**

In this article, we will delve into the world of mathematics and explore the concept of cube roots and their properties. We will examine the given equation $\sqrt[3]{24} \cdot \sqrt[3]{45}$ and determine the correct product. This equation may seem daunting at first, but with a step-by-step approach, we can break it down and find the solution.

Before we dive into the equation, let's take a moment to understand what cube roots are. A cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. In mathematical notation, this is represented as $\sqrt[3]{x}$. For example, the cube root of 27 is 3, because $3^3 = 27$.

Now that we have a basic understanding of cube roots, let's break down the given equation $\sqrt[3]{24} \cdot \sqrt[3]{45}$. To do this, we can use the property of cube roots that states $\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}$.

Step 1: Multiply the Numbers Inside the Cube Roots

Using the property mentioned above, we can multiply the numbers inside the cube roots: $24 \cdot 45 = 1080$.

Step 2: Take the Cube Root of the Product

Now that we have the product of the numbers inside the cube roots, we can take the cube root of the result: $\sqrt[3]{1080}$.

Step 3: Simplify the Cube Root

To simplify the cube root, we can look for perfect cubes that divide into 1080. In this case, we can see that $1080 = 10 \cdot 108$, and $108 = 6 \cdot 18$. We can also see that $10 = 2 \cdot 5$ and $18 = 2 \cdot 3^2$. Therefore, we can rewrite the cube root as $\sqrt[3]{2^3 \cdot 3^2 \cdot 5}$.

Step 4: Simplify the Cube Root Further

Using the property of cube roots that states $\sqrt[3]{a^3} = a$, we can simplify the cube root further: $\sqrt[3]{2^3 \cdot 3^2 \cdot 5} = 2 \cdot 3 \cdot \sqrt[3]{5}$.

In conclusion, the product of $\sqrt[3]{24} \cdot \sqrt[3]{45}$ is $2 \cdot 3 \cdot \sqrt[3]{5}$. This can be rewritten as $6(\sqrt[3]{5})$. Therefore, the correct answer is:

C. $6(\sqrt[3]{5}$]

Q: What is the definition of a cube root? A: A cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number.

Q: How do I simplify a cube root? A: To simplify a cube root, look for perfect cubes that divide into the number inside the cube root. Then, use the property of cube roots that states $\sqrt[3]{a^3} = a$ to simplify the cube root further.

Q: What is the product of $\sqrt[3]{24} \cdot \sqrt[3]{45}$? A: The product of $\sqrt[3]{24} \cdot \sqrt[3]{45}$ is $6(\sqrt[3]{5})$.

Q: Why is the correct answer C. $6(\sqrt[3]{5}$]? A: The correct answer is C. $6(\sqrt[3]{5}$] because it is the result of simplifying the cube root of the product of $\sqrt[3]{24} \cdot \sqrt[3]{45}$.

Q: Can I use a calculator to find the cube root of a number? A: Yes, you can use a calculator to find the cube root of a number. However, it's always a good idea to understand the concept of cube roots and how to simplify them manually.