when 4√96 is simplified, the result is a4√b, where a and b are are positive integers, what is a + b?

We are tasked with simplifying \( \sqrt[4]{96} \) into the form \( a \sqrt[4]{b} \), where \( a \) and \( b \) are positive integers. Then, we need to find \( a + b \).

### Step-by-Step Solution:

1. **Prime factorization of 96**:
   \[
   96 = 2^5 \times 3
   \]
   
2. **Express the fourth root**:
   Now, we take the fourth root of \( 96 \):
   \[
   \sqrt[4]{96} = \sqrt[4]{2^5 \times 3}
   \]

3. **Simplify the expression**:
   The fourth root of a product can be written as the product of the fourth roots of the individual factors:
   \[
   \sqrt[4]{2^5 \times 3} = \sqrt[4]{2^5} \times \sqrt[4]{3}
   \]

4. **Break down the powers**:
   We can separate \( 2^5 \) into \( 2^4 \times 2 \) to simplify:
   \[
   \sqrt[4]{2^5} = \sqrt[4]{2^4 \times 2} = \sqrt[4]{2^4} \times \sqrt[4]{2} = 2 \times \sqrt[4]{2}
   \]

5. **Combine the terms**:
   So, we now have:
   \[
   \sqrt[4]{96} = 2 \times \sqrt[4]{2} \times \sqrt[4]{3} = 2 \times \sqrt[4]{6}
   \]

Thus, the simplified form is \( 2 \sqrt[4]{6} \). In this case, \( a = 2 \) and \( b = 6 \).

6. **Find \( a + b \)**:
   \[
   a + b = 2 + 6 = 8
   \]

### Final Answer:
\[
a + b = 8
\]

Would you like further clarification, or details on any specific part?

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Here are some related questions for further practice:

1. Simplify \( \sqrt[4]{128} \) and find \( a + b \).
2. Simplify \( \sqrt[4]{16 \times 81} \).
3. Find the cube root of 64 and express it in the form \( a \sqrt[3]{b} \).
4. Simplify \( \sqrt[4]{625} \).
5. What is \( \sqrt[3]{1000} \) in its simplest form?

**Tip**: Prime factorization is often a useful technique to simplify roots.

When 4√96 is simplified, the result is a4√b, where a and b are positive integers, what is a + b?

Learn how to simplify the fourth root of 96 and express it in the form a4√b. This step-by-step solution covers key mathematical concepts like prime factorization and properties of roots. Suitable for students in Grades 9-11.

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