https://file.aiquickdraw.com/mathbot/file/17296330868441qpqtxyp.jpg

The expression given in the image is:

\[
\sqrt[4]{96t^{12}u^{18}}
\]

We need to simplify this radical expression. Let's break it down step by step.

### Step 1: Prime factorization of 96
We start by simplifying the constants. The number \( 96 \) can be factored as:

\[
96 = 2^5 \times 3
\]

So the expression becomes:

\[
\sqrt[4]{2^5 \times 3 \times t^{12} \times u^{18}}
\]

### Step 2: Apply the 4th root
The 4th root can be applied to each factor separately:

\[
\sqrt[4]{2^5} \times \sqrt[4]{3} \times \sqrt[4]{t^{12}} \times \sqrt[4]{u^{18}}
\]

#### Simplifying each term:
- \( \sqrt[4]{2^5} = 2^{5/4} \)
- \( \sqrt[4]{3} \) remains as \( \sqrt[4]{3} \), since 3 doesn't have a perfect 4th root.
- \( \sqrt[4]{t^{12}} = t^{12/4} = t^3 \)
- \( \sqrt[4]{u^{18}} = u^{18/4} = u^{9/2} = u^{4.5} \)

### Step 3: Combine the simplified terms
Now combine all the simplified terms:

\[
2^{5/4} \times \sqrt[4]{3} \times t^3 \times u^{4.5}
\]

This is the simplified form of the given expression.

Would you like any further clarification or more detailed steps?

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Here are five related questions to expand on this topic:
1. How do you simplify cube roots in a similar expression?
2. What is the process for simplifying higher-order radicals (like 5th or 6th roots)?
3. How would the simplification differ if the variables had fractional exponents to start with?
4. How do you handle radicals involving negative exponents?
5. How can radicals be simplified when there are multiple variables with different exponents?

### Tip:
Always factor constants into their prime factorization when simplifying radicals—it helps break down the problem into smaller, manageable steps.

Write the following expression in simplified radical form: fourth root of (96t^12u^18). Assume that all of the variables in the expression represent positive real numbers.

This problem explains how to simplify the fourth root of the expression 96t^12u^18. Learn how to break down constants using prime factorization and simplify variables with exponents. Suitable for students in Grades 9-12.

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