What Is This Expression In Simplified Form?$\sqrt[3]{-1,024}$A. $-8 \sqrt[3]{16}$ B. -32 C. $-4 \sqrt[3]{2}$ D. $-8 \sqrt[3]{2}$

Feb 26, 2025 by ADMIN 134 views

What is this expression in simplified form? $\sqrt[3]{-1,024}$

Understanding the Problem

The given expression is a cube root of a negative number, which can be simplified using the properties of exponents and radicals. To simplify this expression, we need to find the cube root of -1,024 and then express it in a simplified form.

Breaking Down the Expression

The cube root of a number can be expressed as the number raised to the power of 1/3. In this case, we have $\sqrt[3]{-1,024}$ , which can be written as $(-1,024)^{1/3}$ .

Simplifying the Expression

To simplify the expression, we can start by factoring -1,024 into its prime factors. We can write -1,024 as $-2^7 \times 2^0$ , which simplifies to $-2^7$ .

Using the Properties of Exponents

Now that we have factored -1,024 into its prime factors, we can use the properties of exponents to simplify the expression. We know that $a^m \times a^n = a^{m+n}$ , so we can rewrite $-2^7$ as $-2^{7-0}$ , which simplifies to $-2^7$ .

Applying the Cube Root

Now that we have simplified the expression to $-2^7$ , we can apply the cube root to get $\sqrt[3]{-2^7}$ . Using the property of cube roots, we know that $\sqrt[3]{a^m} = a^{m/3}$ , so we can rewrite $\sqrt[3]{-2^7}$ as $-2^{7/3}$ .

Simplifying the Fractional Exponent

Now that we have applied the cube root, we can simplify the fractional exponent. We know that $a^{m/n} = (a^m)^{1/n}$ , so we can rewrite $-2^{7/3}$ as $(-2^7)^{1/3}$ .

Evaluating the Expression

Now that we have simplified the expression to $(-2^7)^{1/3}$ , we can evaluate it. We know that $2^7 = 128$ , so we can rewrite $(-2^7)^{1/3}$ as $(-128)^{1/3}$ .

Finding the Cube Root

Finally, we can find the cube root of -128. We know that the cube root of a negative number is the negative of the cube root of its absolute value, so we can rewrite $(-128)^{1/3}$ as $-4\sqrt[3]{2}$ .

Conclusion

In conclusion, the simplified form of the expression $\sqrt[3]{-1,024}$ is $-4\sqrt[3]{2}$ . This is the correct answer among the options provided.

Answer

The correct answer is C. $-4 \sqrt[3]{2}$ .

Explanation

The expression $\sqrt[3]{-1,024}$ can be simplified by factoring -1,024 into its prime factors and then applying the cube root. The simplified form of the expression is $-4\sqrt[3]{2}$ , which is the correct answer among the options provided.

Key Takeaways

The cube root of a negative number is the negative of the cube root of its absolute value.
The expression $\sqrt[3]{-1,024}$ can be simplified by factoring -1,024 into its prime factors and then applying the cube root.
The simplified form of the expression is $-4\sqrt[3]{2}$ .

Common Mistakes

Not factoring the number into its prime factors before applying the cube root.
Not using the properties of exponents to simplify the expression.
Not evaluating the expression correctly.

Tips and Tricks

Make sure to factor the number into its prime factors before applying the cube root.
Use the properties of exponents to simplify the expression.
Evaluate the expression correctly by finding the cube root of the absolute value of the number.

Real-World Applications

The cube root of a negative number has applications in physics and engineering, where it is used to describe the behavior of particles and systems.
The expression $\sqrt[3]{-1,024}$ can be used to model real-world problems, such as the motion of a particle or the behavior of a system.

Conclusion

In conclusion, the expression $\sqrt[3]{-1,024}$ can be simplified by factoring -1,024 into its prime factors and then applying the cube root. The simplified form of the expression is $-4\sqrt[3]{2}$ , which is the correct answer among the options provided.
Q&A: Simplifying Cube Roots of Negative Numbers

Q: What is the cube root of a negative number?

A: The cube root of a negative number is the negative of the cube root of its absolute value. In other words, $\sqrt[3]{-a} = -\sqrt[3]{a}$ .

Q: How do I simplify the cube root of a negative number?

A: To simplify the cube root of a negative number, you need to factor the number into its prime factors and then apply the cube root. You can use the properties of exponents to simplify the expression.

Q: What is the simplified form of $\sqrt[3]{-1,024}$ ?

A: The simplified form of $\sqrt[3]{-1,024}$ is $-4\sqrt[3]{2}$ .

Q: Why do I need to factor the number into its prime factors before applying the cube root?

A: Factoring the number into its prime factors helps you to simplify the expression and make it easier to work with. It also allows you to use the properties of exponents to simplify the expression.

Q: What are some common mistakes to avoid when simplifying cube roots of negative numbers?

A: Some common mistakes to avoid include not factoring the number into its prime factors before applying the cube root, not using the properties of exponents to simplify the expression, and not evaluating the expression correctly.

Q: How do I evaluate the cube root of a negative number?

A: To evaluate the cube root of a negative number, you need to find the cube root of the absolute value of the number and then take the negative of that value.

Q: What are some real-world applications of cube roots of negative numbers?

A: Cube roots of negative numbers have applications in physics and engineering, where they are used to describe the behavior of particles and systems. They can also be used to model real-world problems, such as the motion of a particle or the behavior of a system.

Q: Can I use a calculator to simplify cube roots of negative numbers?

A: Yes, you can use a calculator to simplify cube roots of negative numbers. However, it's always a good idea to understand the underlying math and be able to simplify the expression by hand.

Q: How do I know which option is correct when simplifying a cube root of a negative number?

A: To determine which option is correct, you need to simplify the expression by factoring the number into its prime factors and then applying the cube root. You can then compare your answer to the options provided.

Q: What if I get a negative answer when simplifying a cube root of a negative number?

A: If you get a negative answer when simplifying a cube root of a negative number, it's likely that you have made a mistake. Double-check your work and make sure that you have factored the number into its prime factors and applied the cube root correctly.

Q: Can I use cube roots of negative numbers to solve real-world problems?

A: Yes, you can use cube roots of negative numbers to solve real-world problems. They have applications in physics and engineering, and can be used to model real-world problems, such as the motion of a particle or the behavior of a system.

Q: How do I know if a cube root of a negative number is a rational or irrational number?

A: To determine if a cube root of a negative number is a rational or irrational number, you need to simplify the expression and see if it can be expressed as a ratio of integers. If it can be expressed as a ratio of integers, it is a rational number. If it cannot be expressed as a ratio of integers, it is an irrational number.

Q: Can I use cube roots of negative numbers to solve equations?

A: Yes, you can use cube roots of negative numbers to solve equations. They have applications in algebra and can be used to solve equations involving cube roots.

Q: How do I know if a cube root of a negative number is a real or complex number?

A: To determine if a cube root of a negative number is a real or complex number, you need to simplify the expression and see if it can be expressed as a real number. If it can be expressed as a real number, it is a real number. If it cannot be expressed as a real number, it is a complex number.