Let's carefully go through the steps to simplify the given algebraic expression:
1. Original Expression:
[tex]\[
\sqrt{16a^{16} + 9a^{16}}
\][/tex]
2. Combine Like Terms:
Notice that [tex]\(16a^{16}\)[/tex] and [tex]\(9a^{16}\)[/tex] are both terms involving [tex]\(a^{16}\)[/tex]. We can combine them:
[tex]\[
16a^{16} + 9a^{16} = 25a^{16}
\][/tex]
3. Simplify the Radicand:
Now we take the square root of [tex]\(25a^{16}\)[/tex]:
[tex]\[
\sqrt{25a^{16}}
\][/tex]
4. Further Simplification:
Recall that [tex]\(\sqrt{a^b} = a^{b/2}\)[/tex]. Apply this property to [tex]\(25a^{16}\)[/tex]:
[tex]\[
\sqrt{25a^{16}} = \sqrt{25} \cdot \sqrt{a^{16}} = 5 \cdot a^{16/2} = 5a^8
\][/tex]
Thus, the correct simplification of the original expression is:
[tex]\[
\sqrt{16a^{16} + 9a^{16}} = 5a^8
\][/tex]
Evaluation of the Given Solution:
The given steps were:
[tex]\[
\sqrt{16a^{16} + 9a^{16}} = 4a^4 + 3a^4 = 7a^4
\][/tex]
- The original expression [tex]\(\sqrt{16a^{16} + 9a^{16}}\)[/tex] combines correctly to [tex]\(25a^{16}\)[/tex].
- However, the step claimed as [tex]\(= 4a^4 + 3a^4\)[/tex] is incorrect; this skips the proper utilization of the square root and simplification properties.
- The LHS and RHS of these steps are not equivalent, and thus the final conclusion of [tex]\(7a^4\)[/tex] is incorrect.
Therefore, I do not agree with the given solution. The correct simplified expression should be:
[tex]\[
\sqrt{16a^{16} + 9a^{16}} = 5a^8
\][/tex]
