Which Is Equivalent To $\sqrt[3]{8^{\frac{1}{4}} X}$?A. $8^{\frac{3}{4} X}$ B. $\sqrt[7]{8}^x$ C. $\sqrt[12]{8}^x$ D. $8^{\frac{\sqrt[3]{4 X}}{4 X}}$

Understanding the Problem

In this problem, we are given an expression $\sqrt[3]{8^{\frac{1}{4}} x}$ and asked to find its equivalent form. To simplify this expression, we need to apply the properties of exponents and radicals.

Applying Exponent Rules

The first step is to simplify the expression inside the cube root. We can rewrite $8^{\frac{1}{4}}$ as $(2^3)^{\frac{1}{4}}$. Using the power of a power rule, we can simplify this expression as $2^{\frac{3}{4}}$.

(2^3)^{\frac{1}{4}} = 2^{\frac{3}{4}}

Now, we can rewrite the original expression as $\sqrt[3]{2^{\frac{3}{4}} x}$.

Simplifying the Cube Root

The next step is to simplify the cube root. We can rewrite $\sqrt[3]{2^{\frac{3}{4}} x}$ as $2^{\frac{1}{4}} \sqrt[3]{x}$.

\sqrt[3]{2^{\frac{3}{4}} x} = 2^{\frac{1}{4}} \sqrt[3]{x}

Now, we can rewrite the expression as $2^{\frac{1}{4}} x^{\frac{1}{3}}$.

Comparing with the Options

Let's compare the simplified expression with the options given:

A. $8^{\frac{3}{4} x}$ B. $\sqrt[7]{8}^x$ C. $\sqrt[12]{8}^x$ D. $8^{\frac{\sqrt[3]{4 x}}{4 x}}$

We can see that option A is not equivalent to the simplified expression. Option B can be rewritten as $\sqrt[7]{2^3}^x$, which is not equivalent to the simplified expression. Option C can be rewritten as $\sqrt[12]{2^3}^x$, which is not equivalent to the simplified expression. Option D can be rewritten as $8^{\frac{\sqrt[3]{4 x}}{4 x}}$, which is equivalent to the simplified expression.

Conclusion

In conclusion, the equivalent form of the expression $\sqrt[3]{8^{\frac{1}{4}} x}$ is $8^{\frac{\sqrt[3]{4 x}}{4 x}}$.

Final Answer

Q&A: Simplifying Radical Expressions

Q: What is the difference between a radical and an exponent? A: A radical is a symbol used to represent the square root or cube root of a number, while an exponent is a small number that is raised to a power.

Q: How do I simplify a radical expression? A: To simplify a radical expression, you need to apply the properties of exponents and radicals. You can start by simplifying the expression inside the radical, and then use the power of a power rule to simplify the expression.

Q: What is the power of a power rule? A: The power of a power rule states that when you raise a power to a power, you multiply the exponents. For example, $(a^m)^n = a^{mn}$.

Q: How do I simplify a cube root expression? A: To simplify a cube root expression, you can rewrite it as a power of a number. For example, $\sqrt[3]{a} = a^{\frac{1}{3}}$.

Q: What is the difference between a square root and a cube root? A: A square root is a radical that represents the square root of a number, while a cube root is a radical that represents the cube root of a number.

Q: How do I simplify a radical expression with a variable? A: To simplify a radical expression with a variable, you need to apply the properties of exponents and radicals. You can start by simplifying the expression inside the radical, and then use the power of a power rule to simplify the expression.

Q: What is the final answer to the problem $\sqrt[3]{8^{\frac{1}{4}} x}$? A: The final answer to the problem $\sqrt[3]{8^{\frac{1}{4}} x}$ is $8^{\frac{\sqrt[3]{4 x}}{4 x}}$.

Common Mistakes to Avoid

• Not simplifying the expression inside the radical
• Not applying the power of a power rule
• Not rewriting the cube root as a power of a number
• Not simplifying the expression with a variable

Tips and Tricks

• Always start by simplifying the expression inside the radical
• Use the power of a power rule to simplify the expression
• Rewrite the cube root as a power of a number
• Simplify the expression with a variable by applying the properties of exponents and radicals

Conclusion

In conclusion, simplifying radical expressions requires a good understanding of the properties of exponents and radicals. By following the steps outlined in this article, you can simplify radical expressions with ease. Remember to always start by simplifying the expression inside the radical, and then use the power of a power rule to simplify the expression.