Which Expression Is Equivalent To X 10 4 \sqrt[4]{x^{10}} 4 X 10 ​ ?A. X^2\left(\sqrt[4]{x^2}\right ] B. X 2.5 X^{2.5} X 2.5 C. X 3 ( X 4 X^3(\sqrt[4]{x} X 3 ( 4 X ​ ] D. X 5 X^5 X 5

Understanding the Problem

When dealing with exponents and roots, it's essential to understand the properties of each operation to simplify complex expressions. In this problem, we're given the expression $\sqrt[4]{x^{10}}$ and asked to find an equivalent expression from the given options.

Properties of Exponents and Roots

To simplify the given expression, we need to recall the properties of exponents and roots. The property of a root that we'll use here is:

$$\sqrt[n]{a^n} = a$$

This property states that when a number is raised to a power and then taken to a root of the same degree, the result is the original number.

Simplifying the Expression

Using the property mentioned above, we can simplify the expression $\sqrt[4]{x^{10}}$ as follows:

$$\sqrt[4]{x^{10}} = \sqrt[4]{(x^5)^2}$$

Now, we can apply the property of exponents to simplify further:

$$\sqrt[4]{(x^5)^2} = \sqrt[4]{x^{10}} = x^{\frac{10}{4}} = x^{\frac{5}{2}}$$

Comparing with the Options

Now that we have simplified the expression, let's compare it with the given options:

• A. $x^2\left(\sqrt[4]{x^2}\right)$
• B. $x^{2.5}$
• C. $x^3(\sqrt[4]{x})$
• D. $x^5$

From our simplification, we can see that the correct equivalent expression is:

$$x^{\frac{5}{2}} = x^{2.5}$$

Conclusion

In conclusion, the expression equivalent to $\sqrt[4]{x^{10}}$ is $x^{2.5}$. This can be verified by simplifying the original expression using the properties of exponents and roots.

Final Answer

The final answer is B. $x^{2.5}$.

Q: What is the property of a root that is used to simplify the expression $\sqrt[4]{x^{10}}$?

A: The property of a root that is used to simplify the expression $\sqrt[4]{x^{10}}$ is $\sqrt[n]{a^n} = a$. This property states that when a number is raised to a power and then taken to a root of the same degree, the result is the original number.

Q: How do you simplify the expression $\sqrt[4]{x^{10}}$ using the property of exponents?

A: To simplify the expression $\sqrt[4]{x^{10}}$, we can rewrite it as $\sqrt[4]{(x^5)^2}$. Then, we can apply the property of exponents to simplify further: $\sqrt[4]{(x^5)^2} = \sqrt[4]{x^{10}} = x^{\frac{10}{4}} = x^{\frac{5}{2}}$.

Q: What is the equivalent expression for $\sqrt[4]{x^{10}}$?

A: The equivalent expression for $\sqrt[4]{x^{10}}$ is $x^{2.5}$.

Q: How do you compare the simplified expression with the given options?

A: To compare the simplified expression with the given options, we can rewrite the simplified expression as $x^{2.5}$. Then, we can compare it with the given options:

• A. $x^2\left(\sqrt[4]{x^2}\right)$
• B. $x^{2.5}$
• C. $x^3(\sqrt[4]{x})$
• D. $x^5$

Q: What is the final answer to the problem?

A: The final answer to the problem is B. $x^{2.5}$.

Q: What are some common properties of exponents and roots that are used to simplify expressions?

A: Some common properties of exponents and roots that are used to simplify expressions include:

• $\sqrt[n]{a^n} = a$
• $a^m \cdot a^n = a^{m+n}$
• $(a^m)^n = a^{m \cdot n}$
• $\sqrt[n]{a} = a^{\frac{1}{n}}$

Q: How do you use the properties of exponents and roots to simplify complex expressions?

A: To simplify complex expressions, you can use the properties of exponents and roots to break down the expression into smaller parts. Then, you can apply the properties to simplify each part and combine them to get the final simplified expression.

Q: What are some common mistakes to avoid when simplifying expressions using the properties of exponents and roots?

A: Some common mistakes to avoid when simplifying expressions using the properties of exponents and roots include:

• Not applying the properties correctly
• Not simplifying the expression fully
• Not checking the final answer for errors

Q: How do you check the final answer for errors?

A: To check the final answer for errors, you can plug it back into the original expression and simplify it again. If the final answer matches the original expression, then it is correct. If not, then you need to recheck your work and simplify the expression again.