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

Understanding the Problem

The given problem involves finding the quotient of two cube roots. To solve this, we need to apply the properties of exponents and radicals. The expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ can be simplified using the quotient rule for radicals, which states that $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$.

Applying the Quotient Rule for Radicals

Using the quotient rule for radicals, we can rewrite the given expression as:

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

Simplifying the Expression

Now, we can simplify the fraction inside the cube root:

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

So, the expression becomes:

$$\sqrt[3]{3}$$

Evaluating the Cube Root

The cube root of 3 is a number that, when multiplied by itself twice, gives 3. In other words, it is the number that satisfies the equation $x^3 = 3$. This number is denoted by $\sqrt[3]{3}$.

Conclusion

Therefore, the quotient $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$ is equal to $\sqrt[3]{3}$.

Answer

The correct answer is:

• C. $\sqrt[3]{3}$

Explanation

The other options are incorrect because:

• A. $2\sqrt[3]{5}$ is not equal to $\sqrt[3]{3}$.
• B. 3 is not equal to $\sqrt[3]{3}$.
• D. 40 is not equal to $\sqrt[3]{3}$.

Key Takeaways

• The quotient rule for radicals states that $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$.
• To simplify the expression $\frac{\sqrt[3]{60}}{\sqrt[3]{20}}$, we can use the quotient rule for radicals.
• The cube root of 3 is a number that, when multiplied by itself twice, gives 3.

Practice Problems

1. Find the quotient $\frac{\sqrt[3]{72}}{\sqrt[3]{8}}$.
2. Simplify the expression $\frac{\sqrt[3]{108}}{\sqrt[3]{27}}$.
3. Evaluate the cube root of 27.

Solutions

1. $\frac{\sqrt[3]{72}}{\sqrt[3]{8}} = \sqrt[3]{\frac{72}{8}} = \sqrt[3]{9} = 3\sqrt[3]{3}$
2. $\frac{\sqrt[3]{108}}{\sqrt[3]{27}} = \sqrt[3]{\frac{108}{27}} = \sqrt[3]{4} = 2\sqrt[3]{2}$
3. $\sqrt[3]{27} = 3$

Conclusion

Frequently Asked Questions

Q: What is the quotient rule for radicals?

A: The quotient rule for radicals states that $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$. This rule allows us to simplify expressions involving the quotient of two radicals.

Q: How do I apply the quotient rule for radicals?

A: To apply the quotient rule for radicals, simply divide the numbers inside the radicals and keep the same index (or root). For example, $\frac{\sqrt[3]{60}}{\sqrt[3]{20}} = \sqrt[3]{\frac{60}{20}}$.

Q: What is the cube root of 3?

A: The cube root of 3 is a number that, when multiplied by itself twice, gives 3. In other words, it is the number that satisfies the equation $x^3 = 3$. This number is denoted by $\sqrt[3]{3}$.

Q: How do I simplify the expression $\frac{\sqrt[3]{108}}{\sqrt[3]{27}}$?

A: To simplify the expression $\frac{\sqrt[3]{108}}{\sqrt[3]{27}}$, we can use the quotient rule for radicals. First, divide the numbers inside the radicals: $\frac{108}{27} = 4$. Then, take the cube root of the result: $\sqrt[3]{4} = 2\sqrt[3]{2}$.

Q: What is the quotient $\frac{\sqrt[3]{72}}{\sqrt[3]{8}}$?

A: To find the quotient $\frac{\sqrt[3]{72}}{\sqrt[3]{8}}$, we can use the quotient rule for radicals. First, divide the numbers inside the radicals: $\frac{72}{8} = 9$. Then, take the cube root of the result: $\sqrt[3]{9} = 3\sqrt[3]{3}$.

Q: How do I evaluate the cube root of 27?

A: To evaluate the cube root of 27, we can use the fact that $3^3 = 27$. Therefore, $\sqrt[3]{27} = 3$.

Q: What is the quotient rule for radicals in terms of exponents?

A: The quotient rule for radicals can also be expressed in terms of exponents as $\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} = (a/b)^{\frac{1}{n}}$.

Q: How do I simplify the expression $\frac{\sqrt[3]{1000}}{\sqrt[3]{125}}$?

A: To simplify the expression $\frac{\sqrt[3]{1000}}{\sqrt[3]{125}}$, we can use the quotient rule for radicals. First, divide the numbers inside the radicals: $\frac{1000}{125} = 8$. Then, take the cube root of the result: $\sqrt[3]{8} = 2\sqrt[3]{2}$.

Q: What is the quotient $\frac{\sqrt[3]{216}}{\sqrt[3]{8}}$?

A: To find the quotient $\frac{\sqrt[3]{216}}{\sqrt[3]{8}}$, we can use the quotient rule for radicals. First, divide the numbers inside the radicals: $\frac{216}{8} = 27$. Then, take the cube root of the result: $\sqrt[3]{27} = 3$.

Conclusion

In this article, we have answered some frequently asked questions about the quotient of cube roots. We have discussed the quotient rule for radicals, how to apply it, and how to simplify expressions involving the quotient of two radicals. We have also evaluated the cube root of 3 and simplified some practice problems. The key takeaways from this article are the quotient rule for radicals and the properties of cube roots.