Evaluate \[$(- \sqrt[3]{1000})^3\$\].

Feb 26, 2025 by ADMIN 38 views

$Evaluate ${$(- \sqrt[3]{1000})^3\$}$$

Understanding the Problem

When evaluating the expression ${$ (- \sqrt[3]{1000})^3$}$, it's essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). In this case, we have a nested expression that involves a cube root and a cube.

Breaking Down the Expression

To simplify the expression, we need to start from the innermost operation, which is the cube root of 1000. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In mathematical notation, this is represented as ${$ \sqrt[3]{x} = y \Rightarrow y^3 = x$}$.

Evaluating the Cube Root

The cube root of 1000 can be evaluated as follows:

${$ \sqrt[3]{1000} = \sqrt[3]{10^3} = 10$}$

This is because the cube root of a perfect cube is the number that was cubed. In this case, ${ $10^3 = 1000\$}$ .

Applying the Negative Sign

Now that we have the cube root of 1000, we need to apply the negative sign. When a negative number is raised to an odd power, the result is negative. In this case, we have ${$ (- \sqrt[3]{1000})^3$}$, which can be simplified as follows:

${$ (- \sqrt[3]{1000})^3 = (-10)^3 = -1000$}$

Conclusion

In conclusion, the expression ${$ (- \sqrt[3]{1000})^3$}$ evaluates to -1000. This is because the cube root of 1000 is 10, and when raised to the power of 3, the negative sign is preserved.

Real-World Applications

While the expression ${$ (- \sqrt[3]{1000})^3$}$ may seem abstract, it has real-world applications in various fields, such as:

Physics: In physics, the cube root and cube operations are used to describe the behavior of particles and waves. For example, the energy of a particle can be described using the cube root of its mass.
Engineering: In engineering, the cube root and cube operations are used to describe the behavior of complex systems, such as electrical circuits and mechanical systems.
Computer Science: In computer science, the cube root and cube operations are used in algorithms for solving complex problems, such as finding the shortest path in a graph.

Tips and Tricks

When evaluating expressions involving cube roots and cubes, it's essential to follow the order of operations and simplify the expression step by step. Here are some tips and tricks to keep in mind:

Use the properties of exponents: When simplifying expressions involving exponents, use the properties of exponents, such as ${$ a^m \cdot a^n = a^{m+n}$}$ and ${$ (a^m)n = a^{mn}$}$.
Simplify the expression step by step: When simplifying expressions involving cube roots and cubes, simplify the expression step by step, starting from the innermost operation.
Use the properties of negative numbers: When simplifying expressions involving negative numbers, use the properties of negative numbers, such as ${$ (-a)^n = a^n$}$ and ${$ (-a)^m \cdot (-a)^n = (-a)^{m+n}$}$.

Common Mistakes

When evaluating expressions involving cube roots and cubes, it's easy to make mistakes. Here are some common mistakes to avoid:

Not following the order of operations: When simplifying expressions involving cube roots and cubes, it's essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
Not simplifying the expression step by step: When simplifying expressions involving cube roots and cubes, simplify the expression step by step, starting from the innermost operation.
Not using the properties of exponents and negative numbers: When simplifying expressions involving exponents and negative numbers, use the properties of exponents and negative numbers, such as ${$ a^m \cdot a^n = a^{m+n}$}$ and ${$ (-a)^n = a^n$}$.

Final Thoughts

In conclusion, the expression ${$ (- \sqrt[3]{1000})^3$}$ evaluates to -1000. This is because the cube root of 1000 is 10, and when raised to the power of 3, the negative sign is preserved. When simplifying expressions involving cube roots and cubes, it's essential to follow the order of operations and simplify the expression step by step. By following these tips and tricks, you can avoid common mistakes and simplify complex expressions with ease.

Frequently Asked Questions

Q: What is the cube root of 1000?

A: The cube root of 1000 is 10, because ${$ \sqrt[3]{1000} = \sqrt[3]{10^3} = 10$}$.

Q: What happens when a negative number is raised to an odd power?

A: When a negative number is raised to an odd power, the result is negative. For example, ${$ (-10)^3 = -1000$}$.

Q: How do I simplify the expression ${$ (- \sqrt[3]{1000})^3$}$?

A: To simplify the expression ${$ (- \sqrt[3]{1000})^3$}$, follow these steps:

Evaluate the cube root of 1000, which is 10.
Apply the negative sign to the result, which gives -10.
Raise -10 to the power of 3, which gives -1000.

Q: What are some real-world applications of the expression ${$ (- \sqrt[3]{1000})^3$}$?

A: The expression ${$ (- \sqrt[3]{1000})^3$}$ has real-world applications in various fields, such as:

Physics: In physics, the cube root and cube operations are used to describe the behavior of particles and waves.
Engineering: In engineering, the cube root and cube operations are used to describe the behavior of complex systems, such as electrical circuits and mechanical systems.
Computer Science: In computer science, the cube root and cube operations are used in algorithms for solving complex problems, such as finding the shortest path in a graph.

Q: What are some common mistakes to avoid when simplifying the expression ${$ (- \sqrt[3]{1000})^3$}$?

A: Some common mistakes to avoid when simplifying the expression ${$ (- \sqrt[3]{1000})^3$}$ include:

Not following the order of operations: When simplifying expressions involving cube roots and cubes, it's essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
Not simplifying the expression step by step: When simplifying expressions involving cube roots and cubes, simplify the expression step by step, starting from the innermost operation.
Not using the properties of exponents and negative numbers: When simplifying expressions involving exponents and negative numbers, use the properties of exponents and negative numbers, such as ${$ a^m \cdot a^n = a^{m+n}$}$ and ${$ (-a)^n = a^n$}$.

Q: How can I apply the properties of exponents and negative numbers to simplify the expression ${$ (- \sqrt[3]{1000})^3$}$?

A: To apply the properties of exponents and negative numbers to simplify the expression ${$ (- \sqrt[3]{1000})^3$}$, follow these steps:

Use the property of exponents ${$ a^m \cdot a^n = a^{m+n}$}$ to simplify the expression ${$ (- \sqrt[3]{1000})^3$}$.
Use the property of negative numbers ${$ (-a)^n = a^n$}$ to simplify the expression ${$ (- \sqrt[3]{1000})^3$}$.

Q: What is the final answer to the expression ${$ (- \sqrt[3]{1000})^3$}$?

A: The final answer to the expression ${$ (- \sqrt[3]{1000})^3$}$ is -1000.

Additional Resources

For more information on evaluating expressions involving cube roots and cubes, check out the following resources:

Math textbooks: Consult a math textbook for more information on evaluating expressions involving cube roots and cubes.
Online resources: Check out online resources, such as Khan Academy and Mathway, for more information on evaluating expressions involving cube roots and cubes.
Practice problems: Practice evaluating expressions involving cube roots and cubes with online practice problems or worksheets.