Which Of The Following Is A Like Radical To 6 X 2 3 \sqrt[3]{6 X^2} 3 6 X 2 ​ ?A. X ( 6 X 3 X(\sqrt[3]{6 X} X ( 3 6 X ​ ] B. 6\left(\sqrt[3]{x^2}\right ] C. 4\left(\sqrt[3]{6 X^2}\right ] D. X ( 6 3 X(\sqrt[3]{6} X ( 3 6 ​ ]

$$

Understanding Like Radicals

In mathematics, like radicals are expressions that contain the same index and the same radicand. The index of a radical is the number outside the radical sign, and the radicand is the expression inside the radical sign. For example, $\sqrt[3]{6 x^2}$ and $\sqrt[3]{6 x^2}$ are like radicals because they have the same index (3) and the same radicand ($6 x^2$).

Analyzing the Options

Let's analyze each of the options given:

Option A: $x(\sqrt[3]{6 x})$

This option is not a like radical to $\sqrt[3]{6 x^2}$ because it has a different radicand ($6 x$) and a different index (none, since it's not a radical expression).

Option B: $6\left(\sqrt[3]{x^2}\right)$

This option is not a like radical to $\sqrt[3]{6 x^2}$ because it has a different radicand ($x^2$) and a different index (none, since it's not a radical expression).

Option C: $4\left(\sqrt[3]{6 x^2}\right)$

This option is not a like radical to $\sqrt[3]{6 x^2}$ because it has a different coefficient (4) and a different index (none, since it's not a radical expression).

Option D: $x(\sqrt[3]{6})$

This option is not a like radical to $\sqrt[3]{6 x^2}$ because it has a different radicand ($6$) and a different index (none, since it's not a radical expression).

Finding the Correct Answer

However, we can rewrite the original expression $\sqrt[3]{6 x^2}$ as $\sqrt[3]{6} \cdot \sqrt[3]{x^2}$ using the properties of radicals. This gives us $\sqrt[3]{6} \cdot x^{2/3}$.

Rewriting the Options

Let's rewrite each of the options using the properties of radicals:

Option A: $x(\sqrt[3]{6 x})$

This option can be rewritten as $x \cdot \sqrt[3]{6} \cdot \sqrt[3]{x}$.

Option B: $6\left(\sqrt[3]{x^2}\right)$

This option can be rewritten as $6 \cdot x^{2/3}$.

Option C: $4\left(\sqrt[3]{6 x^2}\right)$

This option can be rewritten as $4 \cdot \sqrt[3]{6} \cdot x^{2/3}$.

Option D: $x(\sqrt[3]{6})$

This option can be rewritten as $x \cdot \sqrt[3]{6}$.

Finding the Correct Answer

Now that we have rewritten each of the options, we can see that only one of them is a like radical to $\sqrt[3]{6 x^2}$.

Conclusion

The correct answer is B. $6\left(\sqrt[3]{x^2}\right)$. This option is a like radical to $\sqrt[3]{6 x^2}$ because it has the same index (3) and the same radicand ($x^2$).

Properties of Radicals

Radicals are expressions that contain a number or expression inside a radical sign. The index of a radical is the number outside the radical sign, and the radicand is the expression inside the radical sign. For example, $\sqrt[3]{6 x^2}$ is a radical expression with an index of 3 and a radicand of $6 x^2$.

Rewriting Radicals

Radicals can be rewritten using the properties of radicals. For example, $\sqrt[3]{6 x^2}$ can be rewritten as $\sqrt[3]{6} \cdot \sqrt[3]{x^2}$.

Like Radicals

Like radicals are expressions that contain the same index and the same radicand. For example, $\sqrt[3]{6 x^2}$ and $\sqrt[3]{6 x^2}$ are like radicals because they have the same index (3) and the same radicand ($6 x^2$).

Conclusion

In conclusion, the correct answer is B. $6\left(\sqrt[3]{x^2}\right)$. This option is a like radical to $\sqrt[3]{6 x^2}$ because it has the same index (3) and the same radicand ($x^2$).

Final Answer

The final answer is B. $6\left(\sqrt[3]{x^2}\right)$.

Q: What are like radicals?

A: Like radicals are expressions that contain the same index and the same radicand. For example, $\sqrt[3]{6 x^2}$ and $\sqrt[3]{6 x^2}$ are like radicals because they have the same index (3) and the same radicand ($6 x^2$).

Q: How do I identify like radicals?

A: To identify like radicals, look for the same index and the same radicand in the expressions. For example, $\sqrt[3]{6 x^2}$ and $\sqrt[3]{6 x^2}$ are like radicals because they have the same index (3) and the same radicand ($6 x^2$).

Q: Can I rewrite radicals using the properties of radicals?

A: Yes, radicals can be rewritten using the properties of radicals. For example, $\sqrt[3]{6 x^2}$ can be rewritten as $\sqrt[3]{6} \cdot \sqrt[3]{x^2}$.

Q: What are the properties of radicals?

A: The properties of radicals include:

• The product of two radicals is equal to the radical of the product of the two expressions.
• The quotient of two radicals is equal to the radical of the quotient of the two expressions.
• The power of a radical is equal to the radical of the power of the expression.

Q: How do I simplify radicals?

A: To simplify radicals, look for perfect squares or perfect cubes in the radicand. For example, $\sqrt[3]{6 x^2}$ can be simplified as $\sqrt[3]{6} \cdot x^{2/3}$.

Q: Can I add or subtract like radicals?

A: Yes, like radicals can be added or subtracted. For example, $\sqrt[3]{6 x^2} + \sqrt[3]{6 x^2}$ can be simplified as $2 \cdot \sqrt[3]{6 x^2}$.

Q: What is the difference between a like radical and a unlike radical?

A: A like radical is an expression that contains the same index and the same radicand, while an unlike radical is an expression that contains a different index or a different radicand.

Q: Can I multiply or divide unlike radicals?

A: No, unlike radicals cannot be multiplied or divided. However, you can multiply or divide the coefficients of the unlike radicals.

Q: How do I multiply or divide radicals?

A: To multiply or divide radicals, multiply or divide the coefficients and the radicands separately. For example, $\sqrt[3]{6 x^2} \cdot \sqrt[3]{6 x^2}$ can be simplified as $\sqrt[3]{6 x^2 \cdot 6 x^2}$.

Q: What is the final answer to the original question?

A: The final answer to the original question is B. $6\left(\sqrt[3]{x^2}\right)$. This option is a like radical to $\sqrt[3]{6 x^2}$ because it has the same index (3) and the same radicand ($x^2$).

Conclusion

In conclusion, like radicals are expressions that contain the same index and the same radicand. Radicals can be rewritten using the properties of radicals, and like radicals can be added or subtracted. Unlike radicals cannot be multiplied or divided, but the coefficients can be multiplied or divided.