Given $x \ \textgreater \ 0$, The Expression $\sqrt[3]{x^8}$ Is Equivalent To:Answer

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Understanding the Problem

When dealing with expressions involving exponents and roots, it's essential to simplify them to their most basic form. In this case, we're given the expression $\sqrt[3]{x^8}$ and asked to find its equivalent form. To simplify this expression, we need to understand the properties of exponents and roots.

Properties of Exponents and Roots

Before we dive into simplifying the expression, let's review some key properties of exponents and roots:

  • Exponent Rules: When we have an expression with an exponent, such as $x^m$, we can raise the base (in this case, $x$) to the power of $m$.
  • Root Rules: When we have an expression with a root, such as $\sqrt[n]{x}$, we can raise the base (in this case, $x$) to the power of $\frac{1}{n}$.

Simplifying the Expression

Now that we've reviewed the properties of exponents and roots, let's simplify the expression $\sqrt[3]{x^8}$. We can start by applying the root rule, which states that $\sqrt[n]{x} = x^{\frac{1}{n}}$. In this case, we have $\sqrt[3]{x^8}$, so we can rewrite it as $(x8){\frac{1}{3}}$.

Applying Exponent Rules

Now that we've rewritten the expression as $(x8){\frac{1}{3}}$, we can apply the exponent rule, which states that $(xm)n = x^{mn}$. In this case, we have $(x8){\frac{1}{3}}$, so we can simplify it to $x^{8 \cdot \frac{1}{3}}$.

Simplifying the Exponent

Now that we've simplified the exponent, we can rewrite the expression as $x^{\frac{8}{3}}$. This is the equivalent form of the original expression $\sqrt[3]{x^8}$.

Conclusion

In conclusion, we've simplified the expression $\sqrt[3]{x^8}$ to its equivalent form, $x^{\frac{8}{3}}$. This demonstrates the importance of understanding the properties of exponents and roots in simplifying complex expressions.

Example Use Case

Let's consider an example use case for this expression. Suppose we have a function $f(x) = \sqrt[3]{x^8}$, and we want to find the value of $f(2)$. Using the simplified expression $x^{\frac{8}{3}}$, we can substitute $x = 2$ to get $f(2) = 2^{\frac{8}{3}}$.

Final Answer

The final answer is: x83\boxed{x^{\frac{8}{3}}}

Understanding the Problem

When dealing with expressions involving exponents and roots, it's essential to simplify them to their most basic form. In this case, we're given the expression $\sqrt[3]{x^8}$ and asked to find its equivalent form. To simplify this expression, we need to understand the properties of exponents and roots.

Q&A: Simplifying the Expression

Q: What is the equivalent form of the expression $\sqrt[3]{x^8}$?

A: The equivalent form of the expression $\sqrt[3]{x^8}$ is $x^{\frac{8}{3}}$.

Q: How do we simplify the expression $\sqrt[3]{x^8}$?

A: To simplify the expression $\sqrt[3]{x^8}$, we can apply the root rule, which states that $\sqrt[n]{x} = x^{\frac{1}{n}}$. We can then rewrite the expression as $(x8){\frac{1}{3}}$ and apply the exponent rule, which states that $(xm)n = x^{mn}$.

Q: What is the final answer to the expression $\sqrt[3]{x^8}$?

A: The final answer to the expression $\sqrt[3]{x^8}$ is $x^{\frac{8}{3}}$.

Q: How do we use the simplified expression in a real-world scenario?

A: We can use the simplified expression $x^{\frac{8}{3}}$ in a real-world scenario by substituting a value for $x$. For example, if we have a function $f(x) = \sqrt[3]{x^8}$, we can substitute $x = 2$ to get $f(2) = 2^{\frac{8}{3}}$.

Common Mistakes

When simplifying the expression $\sqrt[3]{x^8}$, there are a few common mistakes to avoid:

  • Not applying the root rule: Failing to apply the root rule can lead to incorrect simplification of the expression.
  • Not applying the exponent rule: Failing to apply the exponent rule can lead to incorrect simplification of the expression.
  • Not simplifying the exponent: Failing to simplify the exponent can lead to an incorrect final answer.

Conclusion

In conclusion, we've simplified the expression $\sqrt[3]{x^8}$ to its equivalent form, $x^{\frac{8}{3}}$. We've also answered some common questions about simplifying the expression and provided examples of how to use the simplified expression in a real-world scenario.

Final Answer

The final answer is: x83\boxed{x^{\frac{8}{3}}}

Additional Resources

For more information on simplifying expressions involving exponents and roots, check out the following resources:

  • Mathway: A online math problem solver that can help you simplify expressions and solve equations.
  • Khan Academy: A free online resource that provides video lessons and practice exercises on algebra and other math topics.
  • Wolfram Alpha: A online calculator that can help you simplify expressions and solve equations.

Practice Problems

Try simplifying the following expressions:

  • x64\sqrt[4]{x^6}

  • x95\sqrt[5]{x^9}

  • x126\sqrt[6]{x^{12}}

Use the simplified expression $x^{\frac{8}{3}}$ as a reference to help you solve these problems.