Simplify \[$(\sqrt[3]{6x})^4\$\].

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Understanding the Problem

When dealing with exponents and roots, it's essential to understand the properties and rules that govern their behavior. In this case, we're given the expression {(\sqrt[3]{6x})^4$}$ and we're asked to simplify it. To start, let's break down the expression and identify the key components.

Properties of Exponents and Roots

Before we dive into simplifying the expression, let's review some key properties of exponents and roots. When we have an expression in the form of {a^m$}$, where {a$}$ is a number and {m$}$ is an exponent, we can raise it to another power using the property {(am)n = a^{mn}$}$. This property allows us to simplify expressions by combining exponents.

Simplifying the Expression

Now that we've reviewed the properties of exponents and roots, let's apply them to simplify the expression {(\sqrt[3]{6x})^4$}$. We can start by recognizing that the expression is in the form of {(am)n$}$, where {a = \sqrt[3]{6x}$}$ and {m = 1$}$ and {n = 4$}$. Using the property {(am)n = a^{mn}$}$, we can simplify the expression as follows:

{(\sqrt[3]{6x})^4 = (\sqrt[3]{6x})^{4 \cdot 1} = (\sqrt[3]{6x})^4$}$

Applying the Power Rule for Roots

Now that we've simplified the expression using the property {(am)n = a^{mn}$}$, let's apply the power rule for roots to further simplify the expression. The power rule for roots states that {\sqrt[n]{a^m} = a^{m/n}$}$. We can apply this rule to the expression {(\sqrt[3]{6x})^4$}$ as follows:

{(\sqrt[3]{6x})^4 = (6x)^{4/3}$}$

Simplifying the Expression Further

Now that we've applied the power rule for roots, let's simplify the expression further. We can start by recognizing that ${6x = 6 \cdot x\$}. Using the property {a^m \cdot a^n = a^{m+n}$}$, we can simplify the expression as follows:

{(6x)^{4/3} = 6^{4/3} \cdot x^{4/3}$}$

Evaluating the Expression

Now that we've simplified the expression, let's evaluate it. We can start by recognizing that ${6^{4/3} = (6^{1/3})^4 = \sqrt[3]{6}^4\$}. Using the property {(am)n = a^{mn}$}$, we can simplify the expression as follows:

${6^{4/3} = \sqrt[3]{6}^4 = (\sqrt[3]{6})^4\$}

Final Answer

In conclusion, the simplified expression is ${6^{4/3} \cdot x^{4/3}\$}. This expression represents the result of raising the cube root of ${6x\$} to the power of ${4\$}.

Key Takeaways

  • When dealing with exponents and roots, it's essential to understand the properties and rules that govern their behavior.
  • The property {(am)n = a^{mn}$}$ allows us to simplify expressions by combining exponents.
  • The power rule for roots states that {\sqrt[n]{a^m} = a^{m/n}$}$.
  • We can simplify expressions by applying the power rule for roots and combining exponents.

Example Use Cases

  • Simplifying expressions with exponents and roots is a crucial skill in mathematics and is used in a variety of applications, including algebra, calculus, and physics.
  • Understanding the properties and rules of exponents and roots is essential for solving problems in these fields.

Conclusion

In this article, we've simplified the expression {(\sqrt[3]{6x})^4$}$ using the properties and rules of exponents and roots. We've applied the property {(am)n = a^{mn}$}$ to simplify the expression and then applied the power rule for roots to further simplify the expression. The final simplified expression is ${6^{4/3} \cdot x^{4/3}\$}. This expression represents the result of raising the cube root of ${6x\$} to the power of ${4\$}.

Frequently Asked Questions

Q: What is the property of exponents that allows us to simplify the expression {(\sqrt[3]{6x})^4$}$?

A: The property of exponents that allows us to simplify the expression {(\sqrt[3]{6x})^4$}$ is {(am)n = a^{mn}$}$. This property allows us to combine exponents and simplify expressions.

Q: How do we apply the power rule for roots to simplify the expression {(\sqrt[3]{6x})^4$}$?

A: To apply the power rule for roots, we recognize that {\sqrt[n]{a^m} = a^{m/n}$}$. We can apply this rule to the expression {(\sqrt[3]{6x})^4$}$ as follows:

{(\sqrt[3]{6x})^4 = (6x)^{4/3}$}$

Q: What is the final simplified expression for {(\sqrt[3]{6x})^4$}$?

A: The final simplified expression for {(\sqrt[3]{6x})^4$}$ is ${6^{4/3} \cdot x^{4/3}\$}.

Q: How do we evaluate the expression ${6^{4/3}\$}?

A: To evaluate the expression ${6^{4/3}\$}, we recognize that ${6^{4/3} = (6^{1/3})^4 = \sqrt[3]{6}^4\$}. Using the property {(am)n = a^{mn}$}$, we can simplify the expression as follows:

${6^{4/3} = \sqrt[3]{6}^4 = (\sqrt[3]{6})^4\$}

Q: What is the significance of the expression ${6^{4/3}\$}?

A: The expression ${6^{4/3}\$} represents the result of raising the cube root of ${6\$} to the power of ${4\$}.

Q: How do we use the property {(am)n = a^{mn}$}$ to simplify expressions?

A: To use the property {(am)n = a^{mn}$}$ to simplify expressions, we recognize that the expression is in the form of {(am)n$}$. We can then apply the property to simplify the expression as follows:

{(am)n = a^{mn}$}$

Q: What is the power rule for roots?

A: The power rule for roots states that {\sqrt[n]{a^m} = a^{m/n}$}$. This rule allows us to simplify expressions by applying the power rule for roots.

Q: How do we apply the power rule for roots to simplify expressions?

A: To apply the power rule for roots, we recognize that the expression is in the form of {\sqrt[n]{a^m}$}$. We can then apply the power rule for roots to simplify the expression as follows:

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

Q: What is the final answer for the expression {(\sqrt[3]{6x})^4$}$?

A: The final answer for the expression {(\sqrt[3]{6x})^4$}$ is ${6^{4/3} \cdot x^{4/3}\$}.

Example Problems

  • Simplify the expression {(\sqrt[3]{2x})^5$}$.
  • Evaluate the expression ${3^{4/3}\$}.
  • Simplify the expression {(\sqrt[3]{4x})^3$}$.

Solutions

  • {(\sqrt[3]{2x})^5 = (2x)^{5/3}$}$
  • ${3^{4/3} = (\sqrt[3]{3})^4\$}
  • {(\sqrt[3]{4x})^3 = (4x)^{3/3} = 4x$}$

Conclusion

In this article, we've provided a Q&A section to help you understand the properties and rules of exponents and roots. We've also provided example problems and solutions to help you practice simplifying expressions. Remember to apply the property {(am)n = a^{mn}$}$ and the power rule for roots to simplify expressions.