Question

simplify: index of 4 √36m^8n^4 where m≥0 , n≥0

Answer 1

To simplify the expression \( \sqrt[4]{36m^8n^4} \), we can break it down as follows:

1. **Identify Components Inside the Root:**
- \( 36 \) can be expressed as \( 6^2 \).
- \( m^8 \) is already a perfect power.
- \( n^4 \) is also a perfect power.

2. **Rewriting under the Fourth Root:**
\[
\sqrt[4]{36m^8n^4} = \sqrt[4]{6^2 \cdot m^8 \cdot n^4}
\]

3. **Applying the Fourth Root:**
- \( \sqrt[4]{6^2} \) simplifies to \( 6^{2/4} = 6^{1/2} = \sqrt{6} \).
- \( \sqrt[4]{m^8} = m^{8/4} = m^2 \).
- \( \sqrt[4]{n^4} = n^{4/4} = n \).

4. **Combining the Results:**
Putting everything together, we have:
\[
\sqrt[4]{36m^8n^4} = \sqrt{6} \cdot m^2 \cdot n
\]

Thus, the simplified expression is:
\[
m^2n\sqrt{6}
\]

This is the final simplified form of the given expression for \( m \geq 0, n \geq 0 \).