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. ${ 8(\sqrt[3]{10})\$}
Understanding the Problem
The given problem involves the multiplication of two cube roots. To solve this, we need to understand the properties of cube roots and how they interact with multiplication.
Properties of Cube Roots
A cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27. The cube root of a number can be represented using the symbol ∛.
Multiplication of Cube Roots
When we multiply two cube roots, we can combine them into a single cube root. This is based on the property that the cube root of a product is equal to the product of the cube roots. Mathematically, this can be represented as:
∛(a) × ∛(b) = ∛(a × b)
Applying the Property to the Given Problem
In the given problem, we have:
∛(24) × ∛(45)
Using the property mentioned above, we can combine the two cube roots into a single cube root:
∛(24 × 45)
Calculating the Product
To calculate the product of 24 and 45, we simply multiply the two numbers:
24 × 45 = 1080
Simplifying the Cube Root
Now that we have the product, we can simplify the cube root:
∛(1080)
To simplify the cube root, we need to find the largest perfect cube that divides 1080. In this case, the largest perfect cube that divides 1080 is 64 (which is 4^3). We can write 1080 as:
1080 = 64 × 17
Simplifying the Cube Root Further
Now that we have factored 1080 into 64 and 17, we can simplify the cube root:
∛(1080) = ∛(64 × 17)
Using the property that the cube root of a product is equal to the product of the cube roots, we can rewrite this as:
∛(64) × ∛(17)
Evaluating the Cube Root
The cube root of 64 is 4, because 4 × 4 × 4 = 64. Therefore, we can evaluate the cube root as:
∛(64) × ∛(17) = 4 × ∛(17)
Simplifying the Cube Root of 17
The cube root of 17 is a non-perfect cube root, which means it cannot be simplified further. Therefore, we can leave it as is:
4 × ∛(17)
Comparing with the Answer Choices
Now that we have simplified the cube root, we can compare it with the answer choices:
A. ∛(69) B. 4(∛(6)) C. 6(∛(5)) D. 8(∛(10))
Conclusion
Based on our calculations, we can see that the correct answer is:
B. 4(∛(6))
This is because 4 × ∛(17) is equivalent to 4 × ∛(6 × 3), which can be rewritten as 4 × ∛(6) × ∛(3). Since ∛(3) is a non-perfect cube root, we can leave it as is. Therefore, the correct answer is 4(∛(6)).
Final Answer
Q: What is the difference between a cube root and a square root?
A: A cube root and a square root are both types of roots, but they differ in the power to which the number is raised. A square root is the number that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 × 4 = 16. A cube root, on the other hand, is the number that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27.
Q: How do I simplify a cube root?
A: To simplify a cube root, you need to find the largest perfect cube that divides the number inside the cube root. For example, if you have ∛(1080), you can simplify it by finding the largest perfect cube that divides 1080, which is 64 (4^3). You can then rewrite 1080 as 64 × 17 and simplify the cube root as ∛(64) × ∛(17), which is equal to 4 × ∛(17).
Q: What is the property of cube roots that allows us to combine them?
A: The property of cube roots that allows us to combine them is that the cube root of a product is equal to the product of the cube roots. Mathematically, this can be represented as:
∛(a) × ∛(b) = ∛(a × b)
This property allows us to combine two or more cube roots into a single cube root.
Q: How do I evaluate a cube root?
A: To evaluate a cube root, you need to find the number that, when multiplied by itself twice, gives the original number. For example, if you have ∛(27), you can evaluate it as 3, because 3 × 3 × 3 = 27.
Q: Can I simplify a cube root that has a non-perfect cube root inside it?
A: No, you cannot simplify a cube root that has a non-perfect cube root inside it. For example, if you have ∛(17), you cannot simplify it further, because 17 is not a perfect cube.
Q: What is the difference between a cube root and a radical?
A: A cube root and a radical are both types of roots, but they differ in the power to which the number is raised. A cube root is the number that, when multiplied by itself twice, gives the original number. A radical, on the other hand, is a more general term that refers to any root, including square roots, cube roots, and higher-order roots.
Q: How do I use cube roots in real-life applications?
A: Cube roots have many real-life applications, including:
- Calculating volumes of cubes and rectangular prisms
- Finding the length of the side of a cube given its volume
- Solving problems involving rates of change and accumulation
- Modeling population growth and decay
Q: Can I use cube roots to solve problems involving fractions?
A: Yes, you can use cube roots to solve problems involving fractions. For example, if you have the expression ∛(1/8), you can simplify it by finding the cube root of the numerator and the denominator separately, and then simplifying the resulting fraction.
Q: What are some common mistakes to avoid when working with cube roots?
A: Some common mistakes to avoid when working with cube roots include:
- Forgetting to simplify the cube root
- Not using the property of cube roots that allows us to combine them
- Not evaluating the cube root correctly
- Not simplifying the cube root when possible
By avoiding these common mistakes, you can ensure that you are working with cube roots correctly and accurately.