What Is The Following Quotient? 60 3 20 3 \frac{\sqrt[3]{60}}{\sqrt[3]{20}} 3 20 ​ 3 60 ​ ​ A. 3 3 \sqrt[3]{3} 3 3 ​ B. 3 C. 2 ( 5 3 2(\sqrt[3]{5} 2 ( 3 5 ​ ] D. 40

Understanding the Quotient of Cube Roots

When dealing with mathematical expressions involving cube roots, it's essential to understand the properties and rules that govern their behavior. In this article, we will delve into the world of cube roots and explore the quotient of two cube roots. Specifically, we will examine the expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ and determine its value.

The Quotient of Cube Roots

The quotient of two cube roots can be simplified using the rule $\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}$. This rule allows us to simplify complex expressions involving cube roots by dividing the numbers inside the cube roots.

Applying the Quotient Rule

To simplify the expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$, we can apply the quotient rule by dividing the numbers inside the cube roots. This gives us:

$$\frac{\sqrt[3]{60}}{\sqrt[3]{20}} = \sqrt[3]{\frac{60}{20}}$$

Simplifying the Fraction

Now that we have applied the quotient rule, we can simplify the fraction inside the cube root. To do this, we can divide the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 60 and 20 is 20.

$$\frac{60}{20} = 3$$

Substituting the Simplified Fraction

Now that we have simplified the fraction, we can substitute it back into the original expression:

$$\sqrt[3]{\frac{60}{20}} = \sqrt[3]{3}$$

Conclusion

In conclusion, the quotient of $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ is $\sqrt[3]{3}$. This result is obtained by applying the quotient rule for cube roots and simplifying the fraction inside the cube root.

Answer

The correct answer is:

• A. $\sqrt[3]{3}$

Why is this the Correct Answer?

This is the correct answer because we applied the quotient rule for cube roots and simplified the fraction inside the cube root. The quotient rule states that $\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}$, and we simplified the fraction $\frac{60}{20}$ to 3. Therefore, the quotient of $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ is indeed $\sqrt[3]{3}$.

What is the Significance of this Result?

This result has significant implications in mathematics, particularly in the field of algebra. The quotient rule for cube roots is a fundamental concept that allows us to simplify complex expressions involving cube roots. By understanding this rule, we can solve a wide range of mathematical problems involving cube roots.

Real-World Applications

The quotient rule for cube roots has numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, cube roots are used to calculate the volume of irregularly shaped objects. In physics, cube roots are used to calculate the energy of particles. In computer science, cube roots are used in algorithms for solving complex mathematical problems.

Conclusion

In conclusion, the quotient of $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ is $\sqrt[3]{3}$. This result is obtained by applying the quotient rule for cube roots and simplifying the fraction inside the cube root. The quotient rule is a fundamental concept in mathematics that has significant implications in various fields. By understanding this rule, we can solve a wide range of mathematical problems involving cube roots.

Final Answer

The final answer is:

• A. $\sqrt[3]{3}$
  Quotient of Cube Roots: A Q&A Article =====================================

Understanding the Quotient of Cube Roots

In our previous article, we explored the quotient of two cube roots and determined that the expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ simplifies to $\sqrt[3]{3}$. In this article, we will answer some frequently asked questions about the quotient of cube roots.

Q: What is the quotient rule for cube roots?

A: The quotient rule for cube roots states that $\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}$. This rule allows us to simplify complex expressions involving cube roots by dividing the numbers inside the cube roots.

Q: How do I apply the quotient rule for cube roots?

A: To apply the quotient rule, simply divide the numbers inside the cube roots. For example, to simplify the expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$, we can divide the numbers inside the cube roots to get $\sqrt[3]{\frac{60}{20}}$.

Q: What is the greatest common divisor (GCD) of 60 and 20?

A: The greatest common divisor (GCD) of 60 and 20 is 20.

Q: How do I simplify the fraction inside the cube root?

A: To simplify the fraction inside the cube root, we can divide the numerator and denominator by their greatest common divisor (GCD). In this case, we can divide 60 and 20 by 20 to get 3.

Q: What is the final answer to the expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$?

A: The final answer to the expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ is $\sqrt[3]{3}$.

Q: What are some real-world applications of the quotient rule for cube roots?

A: The quotient rule for cube roots has numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, cube roots are used to calculate the volume of irregularly shaped objects. In physics, cube roots are used to calculate the energy of particles. In computer science, cube roots are used in algorithms for solving complex mathematical problems.

Q: Can I use the quotient rule for cube roots with negative numbers?

A: Yes, you can use the quotient rule for cube roots with negative numbers. However, you must remember that the cube root of a negative number is also negative.

Q: Can I use the quotient rule for cube roots with fractions?

A: Yes, you can use the quotient rule for cube roots with fractions. However, you must remember to simplify the fraction inside the cube root.

Q: What are some common mistakes to avoid when using the quotient rule for cube roots?

A: Some common mistakes to avoid when using the quotient rule for cube roots include:

• Forgetting to simplify the fraction inside the cube root
• Not using the greatest common divisor (GCD) to simplify the fraction
• Not remembering that the cube root of a negative number is also negative

Conclusion

In conclusion, the quotient of cube roots is a fundamental concept in mathematics that has numerous real-world applications. By understanding the quotient rule for cube roots, we can simplify complex expressions involving cube roots and solve a wide range of mathematical problems.

Final Answer

The final answer is:

• A. $\sqrt[3]{3}$