Sure! Let's go through the steps to simplify the expression [tex]\((8 - \sqrt{x+6})^2\)[/tex].
### Step 1: Write the expression in expanded form
We start with the expression:
[tex]\[ (8 - \sqrt{x+6})^2 \][/tex]
This is of the form [tex]\((a - b)^2\)[/tex], which expands according to the algebraic identity:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
Here [tex]\(a = 8\)[/tex] and [tex]\(b = \sqrt{x+6}\)[/tex].
### Step 2: Apply the identity
Using the identity, we get:
[tex]\[ (8 - \sqrt{x+6})^2 = 8^2 - 2 \cdot 8 \cdot \sqrt{x+6} + (\sqrt{x+6})^2 \][/tex]
### Step 3: Simplify each term
Let's simplify each term individually:
1. [tex]\(8^2 = 64\)[/tex]
2. [tex]\(2 \cdot 8 \cdot \sqrt{x+6} = 16\sqrt{x+6}\)[/tex]
3. [tex]\((\sqrt{x+6})^2 = x + 6\)[/tex]
### Step 4: Combine the simplified terms
Now, we put these simplified terms back into the expression:
[tex]\[ 64 - 16\sqrt{x+6} + x + 6 \][/tex]
### Step 5: Combine like terms
Combine the constant terms [tex]\(64\)[/tex] and [tex]\(6\)[/tex]:
[tex]\[ x + 64 + 6 - 16\sqrt{x+6} = x + 70 - 16\sqrt{x+6} \][/tex]
### Final simplified expression
Thus, the simplified form of the expression [tex]\((8 - \sqrt{x+6})^2\)[/tex] is:
[tex]\[ x + 70 - 16\sqrt{x+6} \][/tex]
So, the final answer is:
[tex]\[ \boxed{x + 70 - 16\sqrt{x + 6}} \][/tex]
