Simplify The Radical. Assume That All Variables Represent Positive Real Numbers.\[$\sqrt[3]{-8 X^6 Y^{12}}\$\]Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. \[$\sqrt[3]{-8 X^6 Y^{12}}

Understanding the Problem

When dealing with radicals, it's essential to understand the properties and rules that govern them. In this article, we will focus on simplifying the radical $\sqrt[3]{-8 x^6 y^{12}}$. We will break down the problem into manageable steps and provide a clear explanation of each step.

Step 1: Factor the Radicand

The first step in simplifying the radical is to factor the radicand, which is the expression inside the radical sign. In this case, the radicand is $-8 x^6 y^{12}$. We can factor out the greatest perfect cube from the radicand.

$\sqrt[3]{-8 x^6 y^{12}} = \sqrt[3]{(-2)^3 (x^3)^2 (y^6)^2}$

Step 2: Simplify the Radical

Now that we have factored the radicand, we can simplify the radical by taking the cube root of each factor.

$\sqrt[3]{(-2)^3 (x^3)^2 (y^6)^2} = (-2) (x^3) (y^6)$

Step 3: Simplify the Expression

The final step is to simplify the expression by combining like terms.

$(-2) (x^3) (y^6) = -2x^3y^6$

Conclusion

In conclusion, simplifying the radical $\sqrt[3]{-8 x^6 y^{12}}$ involves factoring the radicand, simplifying the radical, and simplifying the expression. By following these steps, we can simplify the radical and arrive at the final answer.

Answer

The final answer is: $\boxed{-2x^3y^6}$

Discussion

This problem requires a good understanding of radicals and their properties. The key to simplifying the radical is to factor the radicand and then simplify the radical by taking the cube root of each factor. By following these steps, we can simplify the radical and arrive at the final answer.

Related Topics

• Simplifying radicals with different indices
• Factoring the radicand
• Simplifying expressions with radicals

Example Problems

• Simplify the radical $\sqrt[3]{27 x^9 y^{18}}$
• Simplify the radical $\sqrt[3]{-64 x^6 y^{12}}$
• Simplify the radical $\sqrt[3]{125 x^3 y^6}$

Practice Problems

• Simplify the radical $\sqrt[3]{-27 x^6 y^{12}}$
• Simplify the radical $\sqrt[3]{64 x^9 y^{18}}$
• Simplify the radical $\sqrt[3]{-125 x^3 y^6}$
  Simplify the Radical: Q&A ==========================

Q: What is a radical?

A: A radical is a mathematical expression that involves the square root or cube root of a number or expression. It is denoted by a symbol called the radical sign, which is a horizontal line above the expression.

Q: What is the difference between a square root and a cube root?

A: A square root is the inverse operation of squaring a number, while a cube root is the inverse operation of cubing a number. In other words, the square root of a number is a value that, when multiplied by itself, gives the original number, while the cube root of a number is a value that, when cubed, gives the original number.

Q: How do I simplify a radical?

A: To simplify a radical, you need to follow these steps:

1. Factor the radicand (the expression inside the radical sign).
2. Simplify the radical by taking the square root or cube root of each factor.
3. Simplify the expression by combining like terms.

Q: What is the greatest perfect cube?

A: The greatest perfect cube is the largest perfect cube that divides the radicand. For example, the greatest perfect cube of 24 is 8, because 8 is the largest perfect cube that divides 24.

Q: How do I simplify a radical with a negative number?

A: To simplify a radical with a negative number, you need to follow these steps:

1. Factor the radicand.
2. Simplify the radical by taking the square root or cube root of each factor.
3. Simplify the expression by combining like terms.
4. If the radicand is negative, the radical will also be negative.

Q: Can I simplify a radical with a variable?

A: Yes, you can simplify a radical with a variable. To do this, you need to follow the same steps as before:

1. Factor the radicand.
2. Simplify the radical by taking the square root or cube root of each factor.
3. Simplify the expression by combining like terms.

Q: What is the difference between a radical and an exponent?

A: A radical and an exponent are both used to represent repeated multiplication, but they are used in different ways. A radical is used to represent the square root or cube root of a number, while an exponent is used to represent repeated multiplication of a number.

Q: Can I simplify a radical with a fraction?

A: Yes, you can simplify a radical with a fraction. To do this, you need to follow the same steps as before:

1. Factor the radicand.
2. Simplify the radical by taking the square root or cube root of each factor.
3. Simplify the expression by combining like terms.

Q: What are some common mistakes to avoid when simplifying radicals?

A: Some common mistakes to avoid when simplifying radicals include:

• Not factoring the radicand properly
• Not simplifying the radical by taking the square root or cube root of each factor
• Not simplifying the expression by combining like terms
• Not considering the sign of the radicand

Q: How do I check my work when simplifying radicals?

A: To check your work when simplifying radicals, you need to follow these steps:

1. Multiply the simplified radical by itself to see if it equals the original radicand.
2. Check if the simplified radical is equal to the original radicand.
3. If the simplified radical is not equal to the original radicand, recheck your work and simplify the radical again.

Q: What are some real-world applications of simplifying radicals?

A: Simplifying radicals has many real-world applications, including:

• Calculating distances and heights in geometry and trigonometry
• Solving problems in physics and engineering
• Calculating areas and volumes in geometry and calculus
• Solving problems in finance and economics