Which Is Equivalent To $(\sqrt[3]{125})^x$?A. $125^{\frac{1}{3} X}$ B. $125^{\frac{1}{3x}}$ C. $125^{3x}$ D. $125^{\left(\frac{1}{3}\right)^x}$

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 $(\sqrt[3]{125})^x$ and determining its equivalent form. We will explore the properties of exponents and radicals, and apply them to simplify the given expression.

Understanding Exponents and Radicals

Before we dive into simplifying the expression, let's review the basics of exponents and radicals.

• Exponents: An exponent is a small number that is written to the upper right of a number or a variable. It represents the number of times the base is multiplied by itself. For example, $a^b$ means $a$ multiplied by itself $b$ times.
• Radicals: A radical is a mathematical expression that involves a root or a power of a number. The most common radical is the square root, denoted by $\sqrt{x}$, which represents the number that, when multiplied by itself, gives the original number $x$. Other common radicals include cube roots, fourth roots, and so on.

Simplifying the Expression

Now that we have a basic understanding of exponents and radicals, let's simplify the expression $(\sqrt[3]{125})^x$.

We can start by evaluating the cube root of 125. Since $5^3 = 125$, we can rewrite the expression as:

$(\sqrt[3]{125})^x = (5)^x$

Next, we can apply the property of exponents that states $(a^b)^c = a^{bc}$. In this case, we have:

$(5)^x = 5^{x}$

Now, we can rewrite the expression in terms of the original base, 125, by applying the property of exponents that states $a^{bc} = (a^b)^c$. In this case, we have:

$5^{x} = (5^1)^x = 125^{\frac{x}{3}}$

However, we are not done yet. We need to find an equivalent expression that matches one of the options provided. Let's analyze each option and see which one matches our simplified expression.

Analyzing the Options

Let's analyze each option and see which one matches our simplified expression.

Option A: $125^{\frac{1}{3} x}$

This option does not match our simplified expression. The exponent is $\frac{1}{3} x$, whereas our simplified expression has an exponent of $\frac{x}{3}$.

Option B: $125^{\frac{1}{3x}}$

This option also does not match our simplified expression. The exponent is $\frac{1}{3x}$, whereas our simplified expression has an exponent of $\frac{x}{3}$.

Option C: $125^{3x}$

This option does not match our simplified expression. The exponent is $3x$, whereas our simplified expression has an exponent of $\frac{x}{3}$.

Option D: $125^{\left(\frac{1}{3}\right)^x}$

This option does not match our simplified expression. The exponent is $\left(\frac{1}{3}\right)^x$, whereas our simplified expression has an exponent of $\frac{x}{3}$.

Conclusion

In conclusion, none of the options provided match our simplified expression. However, we can rewrite the expression in a form that matches one of the options. Let's rewrite the expression as:

$125^{\frac{x}{3}} = (125^{\frac{1}{3}})^x = (5)^x$

This expression matches option A, but with a slight modification. The exponent is $\frac{x}{3}$, whereas the exponent in option A is $\frac{1}{3} x$. However, we can rewrite the expression in option A as:

$125^{\frac{1}{3} x} = (125^{\frac{1}{3}})^x = (5)^x$

Therefore, the correct answer is option A, but with a slight modification.

Final Answer

The final answer is:

$125^{\frac{x}{3}}$

However, we can rewrite the expression in a form that matches one of the options. Let's rewrite the expression as:

$125^{\frac{x}{3}} = (125^{\frac{1}{3}})^x = (5)^x$

This expression matches option A, but with a slight modification. The exponent is $\frac{x}{3}$, whereas the exponent in option A is $\frac{1}{3} x$. However, we can rewrite the expression in option A as:

$125^{\frac{1}{3} x} = (125^{\frac{1}{3}})^x = (5)^x$

Therefore, the correct answer is option A, but with a slight modification.

Additional Tips and Tricks

Here are some additional tips and tricks to help you simplify radical expressions:

• Use the property of exponents: The property of exponents states that $(a^b)^c = a^{bc}$. This property can be used to simplify radical expressions.
• Use the property of radicals: The property of radicals states that $\sqrt[n]{a^n} = a$. This property can be used to simplify radical expressions.
• Use the property of equality: The property of equality states that if $a = b$, then $a^c = b^c$. This property can be used to simplify radical expressions.

By following these tips and tricks, you can simplify radical expressions and solve mathematical problems with ease.

Conclusion

Q&A: Simplifying Radical Expressions

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

A: A radical is a mathematical expression that involves a root or a power of a number. The most common radical is the square root, denoted by $\sqrt{x}$, which represents the number that, when multiplied by itself, gives the original number $x$. An exponent, on the other hand, is a small number that is written to the upper right of a number or a variable. It represents the number of times the base is multiplied by itself.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the following steps:

1. Evaluate the radical expression by finding the root of the number.
2. Apply the property of exponents that states $(a^b)^c = a^{bc}$.
3. Simplify the expression by combining like terms.

Q: What is the property of exponents that states $(a^b)^c = a^{bc}$?

A: The property of exponents that states $(a^b)^c = a^{bc}$ is a fundamental property of exponents. It states that when you raise a power to a power, you can multiply the exponents. For example, $(a^b)^c = a^{bc}$.

Q: How do I use the property of radicals that states $\sqrt[n]{a^n} = a$?

A: To use the property of radicals that states $\sqrt[n]{a^n} = a$, you can follow these steps:

1. Identify the radical expression that you want to simplify.
2. Check if the radicand is a perfect power of the index.
3. If the radicand is a perfect power of the index, you can simplify the radical expression by removing the radical sign.

Q: What is the property of equality that states if $a = b$, then $a^c = b^c$?

A: The property of equality that states if $a = b$, then $a^c = b^c$ is a fundamental property of equality. It states that if two expressions are equal, then their powers are also equal. For example, if $a = b$, then $a^c = b^c$.

Q: How do I use the property of equality to simplify radical expressions?

A: To use the property of equality to simplify radical expressions, you can follow these steps:

1. Identify the radical expression that you want to simplify.
2. Check if the radical expression is equal to another expression.
3. If the radical expression is equal to another expression, you can simplify the radical expression by using the property of equality.

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

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

• Not evaluating the radical expression correctly.
• Not applying the property of exponents correctly.
• Not simplifying the expression by combining like terms.
• Not using the property of radicals correctly.
• Not using the property of equality correctly.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by:

• Working on math problems that involve simplifying radical expressions.
• Using online resources, such as math websites and apps, to practice simplifying radical expressions.
• Asking a teacher or tutor for help with simplifying radical expressions.
• Joining a study group to practice simplifying radical expressions with others.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for solving mathematical problems. By understanding the properties of exponents and radicals, and applying them to simplify radical expressions, you can solve mathematical problems with ease. Remember to use the property of exponents, the property of radicals, and the property of equality to simplify radical expressions. With practice and patience, you can become proficient in simplifying radical expressions and solving mathematical problems.