Select The Correct Answer.Which Expression Is Equivalent To 54 X 3 + 3 2 X 3 \sqrt[3]{54 X}+3 \sqrt[3]{2 X} 3 54 X ​ + 3 3 2 X ​ , If X ≠ 0 X \neq 0 X  = 0 ?A. 6 2 X 3 6 \sqrt[3]{2 X} 6 3 2 X ​ B. 3 56 X 3 3 \sqrt[3]{56 X} 3 3 56 X ​ C. 4 56 X 3 4 \sqrt[3]{56 X} 4 3 56 X ​ D. 5 5 X 3 5 \sqrt[3]{5 X} 5 3 5 X ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given problem: $\sqrt[3]{54 x}+3 \sqrt[3]{2 x}$. We will break down the solution into manageable steps, using algebraic manipulations and properties of radicals to arrive at the correct answer.

Understanding the Problem

The given expression is $\sqrt[3]{54 x}+3 \sqrt[3]{2 x}$. Our goal is to simplify this expression and find an equivalent form. To do this, we need to understand the properties of radicals and how to manipulate them.

Properties of Radicals

Before we dive into the solution, let's review some key properties of radicals:

• Product of Radicals: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$
• Power of a Radical: $(\sqrt[n]{a})^m = \sqrt[n]{a^m}$
• Sum of Radicals: $\sqrt[n]{a} + \sqrt[n]{b} \neq \sqrt[n]{a+b}$ (in general)

Step 1: Factor the Radicand

The first step in simplifying the given expression is to factor the radicand. We can factor $54x$ as $2 \cdot 3^3 \cdot x$. This gives us:

$$\sqrt[3]{54x} = \sqrt[3]{2 \cdot 3^3 \cdot x}$$

Step 2: Simplify the First Radical

Now that we have factored the radicand, we can simplify the first radical:

$$\sqrt[3]{2 \cdot 3^3 \cdot x} = 3\sqrt[3]{2x}$$

Step 3: Simplify the Second Radical

The second radical is already simplified, so we can leave it as is:

$$3\sqrt[3]{2x}$$

Step 4: Combine the Radicals

Now that we have simplified both radicals, we can combine them:

$$\sqrt[3]{54x}+3\sqrt[3]{2x} = 3\sqrt[3]{2x} + 3\sqrt[3]{2x}$$

Step 5: Factor Out the Common Term

We can factor out the common term $3\sqrt[3]{2x}$:

$$3\sqrt[3]{2x} + 3\sqrt[3]{2x} = 6\sqrt[3]{2x}$$

Conclusion

In conclusion, the given expression $\sqrt[3]{54x}+3\sqrt[3]{2x}$ is equivalent to $6\sqrt[3]{2x}$. This is the correct answer, and we arrived at it by simplifying the radical expressions using algebraic manipulations and properties of radicals.

Answer

The correct answer is:

• A. $6 \sqrt[3]{2 x}$

Discussion

This problem requires a deep understanding of radical expressions and their properties. The key to solving this problem is to factor the radicand and simplify the radicals using algebraic manipulations. By following the steps outlined in this article, you should be able to arrive at the correct answer.

Additional Resources

If you are struggling with radical expressions or need additional practice, here are some additional resources:

• Khan Academy: Radical Expressions
• Mathway: Radical Expressions
• Wolfram Alpha: Radical Expressions

Final Thoughts

Q: What is the difference between a radical expression and a polynomial expression?

A: A radical expression is an expression that contains a radical, such as $\sqrt[n]{a}$, where $n$ is a positive integer and $a$ is a real number. A polynomial expression, on the other hand, is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

1. Factor the radicand (the expression inside the radical).
2. Simplify the radical by taking out any perfect squares or cubes.
3. Combine the simplified radicals using the properties of radicals.

Q: What is the property of radicals that allows me to simplify $\sqrt[n]{a} \cdot \sqrt[n]{b}$?

A: The property of radicals that allows you to simplify $\sqrt[n]{a} \cdot \sqrt[n]{b}$ is the product of radicals, which states that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.

Q: Can I simplify $\sqrt[n]{a} + \sqrt[n]{b}$?

A: In general, no, you cannot simplify $\sqrt[n]{a} + \sqrt[n]{b}$. However, you can simplify expressions of the form $\sqrt[n]{a} + \sqrt[n]{b}$ if $n$ is even and $a$ and $b$ are both perfect squares.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you need to follow these steps:

1. Simplify each radical individually using the steps outlined above.
2. Combine the simplified radicals using the properties of radicals.

Q: What is the difference between a rational exponent and a radical exponent?

A: A rational exponent is an exponent that is a fraction, such as $a^{1/2}$ or $a^{3/4}$. A radical exponent, on the other hand, is an exponent that is a root, such as $\sqrt[n]{a}$.

Q: Can I simplify an expression with a rational exponent?

A: Yes, you can simplify an expression with a rational exponent by rewriting it as a radical expression and then simplifying the radical expression using the steps outlined above.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to follow these steps:

1. Rewrite the expression with a positive exponent by taking the reciprocal of the base.
2. Simplify the expression using the steps outlined above.

Q: What is the difference between a radical expression and an algebraic expression?

A: A radical expression is an expression that contains a radical, such as $\sqrt[n]{a}$, where $n$ is a positive integer and $a$ is a real number. An algebraic expression, on the other hand, is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can I simplify an expression with a radical and an algebraic expression?

A: Yes, you can simplify an expression with a radical and an algebraic expression by following the steps outlined above.

Conclusion

Simplifying radical expressions is a crucial skill in mathematics, and it requires a deep understanding of algebraic manipulations and properties of radicals. By following the steps outlined in this article, you should be able to simplify radical expressions and arrive at the correct answer. Remember to practice regularly and seek additional resources if you need help.

Additional Resources

If you are struggling with radical expressions or need additional practice, here are some additional resources:

• Khan Academy: Radical Expressions
• Mathway: Radical Expressions
• Wolfram Alpha: Radical Expressions

Final Thoughts

Simplifying radical expressions is a crucial skill in mathematics, and it requires a deep understanding of algebraic manipulations and properties of radicals. By following the steps outlined in this article, you should be able to simplify radical expressions and arrive at the correct answer. Remember to practice regularly and seek additional resources if you need help.