Evaluate.
 
 
\[\dfrac{3^{^{\scriptsize\dfrac{1}5}}}{\sqrt[5]{-96}}=\]

We are tasked with evaluating the expression:

\[
\frac{3^{\frac{1}{5}}}{\sqrt[5]{-96}}
\]

### Step 1: Simplify the denominator
The denominator is a fifth root of \(-96\). The fifth root of a negative number is also negative, since odd roots of negative numbers are negative. Hence, we calculate:

\[
\sqrt[5]{-96} = -\sqrt[5]{96}
\]

Now we need to approximate the value of \(\sqrt[5]{96}\). Using a calculator, we find:

\[
\sqrt[5]{96} \approx 2.6265
\]

Thus,

\[
\sqrt[5]{-96} \approx -2.6265
\]

### Step 2: Simplify the numerator
The numerator is \(3^{\frac{1}{5}}\). We calculate this value:

\[
3^{\frac{1}{5}} \approx 1.2457
\]

### Step 3: Combine the two
Now, divide the values from the numerator and denominator:

\[
\frac{1.2457}{-2.6265} \approx -0.4744
\]

### Final Answer:
\[
\frac{3^{\frac{1}{5}}}{\sqrt[5]{-96}} \approx -0.4744
\]

Would you like to explore more about this solution or need further details? Here are five related questions to explore:

1. How are negative numbers handled under different types of roots (odd vs. even)?
2. How do you approximate the value of higher roots (e.g., cube roots or fifth roots)?
3. Why do odd roots of negative numbers remain negative?
4. What are other methods to evaluate fractional exponents without a calculator?
5. How can we express roots in terms of exponents for simplification?

**Tip:** When calculating roots of negative numbers, remember that odd roots produce negative results, while even roots are undefined for negative real numbers.

Evaluate. \[\dfrac{3^{\frac{1}{5}}}{\sqrt[5]{-96}}=\]

Explore how to evaluate the expression 3^(1/5) divided by the fifth root of -96. This problem involves exponents and radicals, showcasing how negative numbers behave under odd roots. The step-by-step solution simplifies the problem using approximate values for higher roots. Suitable for students in Grades 10-12.

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