Which Expression Is Equivalent To $(27 X)^{\frac{1}{3}}$?A. 27 X 3 27 \sqrt[3]{x} 27 3 X ​ B. 3 X 3 3 \sqrt[3]{x} 3 3 X ​ C. 9 X 3 9 \sqrt[3]{x} 9 3 X ​ D. 9 X 9 X 9 X

Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $(27 x)^{\frac{1}{3}}$ and determine which of the given options is equivalent to it.

Understanding the Expression

The given expression is $(27 x)^{\frac{1}{3}}$. To simplify this expression, we need to apply the rules of exponents and radicals. The expression can be broken down into two parts: the coefficient $27$ and the variable $x$.

Simplifying the Coefficient

The coefficient $27$ can be expressed as $3^3$. Using the property of exponents, we can rewrite the expression as $(3^3 x)^{\frac{1}{3}}$.

Applying the Power Rule

The power rule states that for any numbers $a$ and $b$ and any integer $n$, $(ab)^n = a^nb^n$. We can apply this rule to the expression $(3^3 x)^{\frac{1}{3}}$ to simplify it further.

$$(3^3 x)^{\frac{1}{3}} = 3^{3 \cdot \frac{1}{3}} x^{\frac{1}{3}}$$

Simplifying the Exponent

The exponent $3 \cdot \frac{1}{3}$ can be simplified to $1$. Therefore, the expression becomes:

$$3^1 x^{\frac{1}{3}}$$

Simplifying the Radical

The expression $x^{\frac{1}{3}}$ can be rewritten as $\sqrt[3]{x}$. Therefore, the final simplified expression is:

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

Conclusion

Based on the simplification process, we can conclude that the expression $(27 x)^{\frac{1}{3}}$ is equivalent to $3 \sqrt[3]{x}$.

Discussion

The correct answer is option B: $3 \sqrt[3]{x}$. This is because the simplified expression $3 \sqrt[3]{x}$ is equivalent to the original expression $(27 x)^{\frac{1}{3}}$.

Comparison with Other Options

Let's compare the simplified expression $3 \sqrt[3]{x}$ with the other options:

• Option A: $27 \sqrt[3]{x}$ is not equivalent to the original expression because the coefficient is incorrect.
• Option C: $9 \sqrt[3]{x}$ is not equivalent to the original expression because the coefficient is incorrect.
• Option D: $9 x$ is not equivalent to the original expression because it does not contain the radical term.

Conclusion

In conclusion, the expression $(27 x)^{\frac{1}{3}}$ is equivalent to $3 \sqrt[3]{x}$. This is because the simplified expression $3 \sqrt[3]{x}$ is equivalent to the original expression. The other options are not equivalent to the original expression.

Final Answer

The final answer is option B: $3 \sqrt[3]{x}$.

Additional Tips and Tricks

• When simplifying radical expressions, it's essential to apply the rules of exponents and radicals correctly.
• The power rule states that for any numbers $a$ and $b$ and any integer $n$, $(ab)^n = a^nb^n$.
• The exponent rule states that for any numbers $a$ and $b$ and any integer $n$, $(a^m)^n = a^{mn}$.
• When simplifying radical expressions, it's essential to rewrite the expression in a simplified form.

Common Mistakes to Avoid

• Not applying the rules of exponents and radicals correctly.
• Not simplifying the expression correctly.
• Not rewriting the expression in a simplified form.

Conclusion

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Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will provide a Q&A guide to help you understand how to simplify radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a radical sign, which is denoted by the symbol $\sqrt[n]{x}$, where $n$ is a positive integer.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to apply the rules of exponents and radicals. The first step is to identify the coefficient and the variable in the expression. Then, you can apply the power rule and the exponent rule to simplify the expression.

Q: What is the power rule?

A: The power rule states that for any numbers $a$ and $b$ and any integer $n$, $(ab)^n = a^nb^n$. This rule allows you to simplify expressions by separating the coefficient and the variable.

Q: What is the exponent rule?

A: The exponent rule states that for any numbers $a$ and $b$ and any integer $n$, $(a^m)^n = a^{mn}$. This rule allows you to simplify expressions by combining the exponents.

Q: How do I simplify an expression with a coefficient and a variable?

A: To simplify an expression with a coefficient and a variable, you need to apply the power rule and the exponent rule. For example, if you have the expression $(3x)^2$, you can simplify it by applying the power rule:

$$(3x)^2 = 3^2 x^2 = 9x^2$$

Q: How do I simplify an expression with a radical sign?

A: To simplify an expression with a radical sign, you need to apply the rules of radicals. For example, if you have the expression $\sqrt[3]{27x}$, you can simplify it by applying the rule of radicals:

$$\sqrt[3]{27x} = \sqrt[3]{3^3 x} = 3 \sqrt[3]{x}$$

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not applying the rules of exponents and radicals correctly
• Not simplifying the expression correctly
• Not rewriting the expression in a simplified form

Q: How do I determine which option is equivalent to the original expression?

A: To determine which option is equivalent to the original expression, you need to simplify the expression and compare it with the options. You can use the rules of exponents and radicals to simplify the expression and then compare it with the options.

Q: What are some tips and tricks for simplifying radical expressions?

A: Some tips and tricks for simplifying radical expressions include:

• Applying the rules of exponents and radicals correctly
• Simplifying the expression correctly
• Rewriting the expression in a simplified form
• Using the power rule and the exponent rule to simplify expressions

Conclusion

In conclusion, simplifying radical expressions is a crucial concept in mathematics. By applying the rules of exponents and radicals correctly, you can simplify radical expressions and determine which option is equivalent to the original expression. Remember to avoid common mistakes and use the tips and tricks provided in this article to help you simplify radical expressions.

Final Tips and Tricks

• Always apply the rules of exponents and radicals correctly
• Simplify the expression correctly
• Rewrite the expression in a simplified form
• Use the power rule and the exponent rule to simplify expressions
• Avoid common mistakes when simplifying radical expressions

Common Mistakes to Avoid

• Not applying the rules of exponents and radicals correctly
• Not simplifying the expression correctly
• Not rewriting the expression in a simplified form

Conclusion

In conclusion, simplifying radical expressions is a crucial concept in mathematics. By applying the rules of exponents and radicals correctly, you can simplify radical expressions and determine which option is equivalent to the original expression. Remember to avoid common mistakes and use the tips and tricks provided in this article to help you simplify radical expressions.