Sure, let's carefully analyze and simplify the given algebraic expression step-by-step to see if Sam's solution is correct and to redo the calculations if necessary.
Expression:
[tex]\[ \sqrt{16a^{16} + 9a^{16}} \][/tex]
### Step-by-Step Simplification:
1. Combine Like Terms Inside the Square Root:
[tex]\[
16a^{16} + 9a^{16} = (16 + 9)a^{16} = 25a^{16}
\][/tex]
2. Simplify the Expression Under the Square Root:
[tex]\[
25a^{16}
\][/tex]
3. Apply the Square Root to the Simplified Expression:
[tex]\[
\sqrt{25a^{16}}
\][/tex]
4. Break Down Further:
[tex]\[
\sqrt{25} \cdot \sqrt{a^{16}}
\][/tex]
5. Simplify Each Component:
[tex]\[
\sqrt{25} = 5
\][/tex]
[tex]\[
\sqrt{a^{16}} = a^{8}
\][/tex]
6. Combine the Results:
[tex]\[
5 \cdot a^{8} = 5a^{8}
\][/tex]
Therefore, the correct simplification is:
[tex]\[ \sqrt{16a^{16} + 9a^{16}} = 5a^{8} \][/tex]
### Comparison with Sam's Solution:
- Sam's attempted simplification was:
[tex]\[
\sqrt{16a^{16} + 9a^{16}} = 4a^4 + 3a^4 = 7a^4
\][/tex]
### Error Analysis:
- Sam incorrectly simplified the expression by adding the coefficients inside the square root directly and then combining them in a manner that does not respect the properties of square roots.
- Proper simplification shows that the correct final expression should be [tex]\( 5a^{8} \)[/tex].
In conclusion, I do not agree with Sam's solution. The correct simplification of the expression [tex]\(\sqrt{16a^{16} + 9a^{16}}\)[/tex] is [tex]\(5a^{8}\)[/tex].
