To simplify the expression [tex]\( 4 \sqrt{16 a^{16}} + 9 a^{16} \)[/tex], we will break it down step by step.
1. Simplify the Square Root Expression:
- Start with the term inside the square root: [tex]\( 16 a^{16} \)[/tex].
- We recognize that [tex]\( 16 \)[/tex] is a perfect square: [tex]\( 16 = 4^2 \)[/tex].
- We also recognize that [tex]\( a^{16} \)[/tex] can be simplified as [tex]\( (a^8)^2 \)[/tex].
Therefore, [tex]\( 16 a^{16} = (4a^8)^2 \)[/tex].
- The square root of [tex]\( (4a^8)^2 \)[/tex] is:
[tex]\[
\sqrt{(4a^8)^2} = 4a^8
\][/tex]
2. Multiply by the Coefficient Outside the Square Root:
- We then multiply the simplified square root result by the coefficient outside, which is [tex]\( 4 \)[/tex]:
[tex]\[
4 \cdot 4a^8 = 16a^8
\][/tex]
3. Add the Remaining Term:
- Now we combine the simplified term with the remaining term [tex]\( 9a^{16} \)[/tex]:
[tex]\[
16a^8 + 9a^{16}
\][/tex]
So the simplified expression is:
[tex]\[
9a^{16} + 16a^8
\][/tex]
This is the final simplified form of the given expression [tex]\( 4 \sqrt{16 a^{16}} + 9 a^{16} \)[/tex].
