Let's analyze and simplify the given algebraic expression step-by-step:
The original expression is:
[tex]\[ \sqrt{16a^{16} + 9a^{16}} \][/tex]
First, we combine like terms inside the square root:
[tex]\[ 16a^{16} + 9a^{16} = (16 + 9)a^{16} = 25a^{16} \][/tex]
Next, we can simplify the square root of the combined term:
[tex]\[ \sqrt{25a^{16}} \][/tex]
We know that the square root of a product is the product of the square roots:
[tex]\[ \sqrt{25a^{16}} = \sqrt{25} \times \sqrt{a^{16}} \][/tex]
Now, we find the square roots of each of the components:
[tex]\[ \sqrt{25} = 5 \][/tex]
[tex]\[ \sqrt{a^{16}} = a^8 \][/tex]
(Recall that [tex]\(\sqrt{a^{16}} = a^{16/2} = a^8\)[/tex])
Putting it all together, we get:
[tex]\[ \sqrt{25a^{16}} = 5a^8 \][/tex]
Therefore, Sam's given answer of [tex]\( 7a^4 \)[/tex] is incorrect. The correct simplified form of the expression is:
[tex]\[ 5a^8 \][/tex]
