To determine if Sam's simplification is correct, let's carefully examine the given expression and simplify it step by step:
Given expression:
[tex]\[ \sqrt{16a^{16} + 9a^{16}} \][/tex]
Step 1: Combine the terms inside the square root.
[tex]\[ 16a^{16} + 9a^{16} = (16 + 9)a^{16} = 25a^{16} \][/tex]
Step 2: Simplify the square root.
[tex]\[ \sqrt{25a^{16}} \][/tex]
Step 3: Apply the property of square roots to separate the coefficient and the variable.
[tex]\[ \sqrt{25a^{16}} = \sqrt{25} \times \sqrt{a^{16}} \][/tex]
Step 4: Simplify each part separately.
[tex]\[ \sqrt{25} = 5 \][/tex]
[tex]\[ \sqrt{a^{16}} = a^{8} \][/tex]
Step 5: Multiply the simplified parts.
[tex]\[ \sqrt{25a^{16}} = 5 \times a^{8} = 5a^{8} \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ 5a^{8} \][/tex]
Now, let’s compare this with Sam's result:
- Sam obtained:
[tex]\[ 7a^{4} \][/tex]
Comparing [tex]\(5a^{8}\)[/tex] with [tex]\(7a^{4}\)[/tex], they are clearly different. Therefore, Sam's result is incorrect.
The correct simplified result for [tex]\( \sqrt{16a^{16} + 9a^{16}} \)[/tex] is:
[tex]\[ 5a^{8} \][/tex]
