Sure! Let's demonstrate that [tex]\((6 + 2 \sqrt{12})^2 = 12(7 + 4 \sqrt{3})\)[/tex] step by step.
### Step 1: Simplify the Expression Inside the Parentheses
First, look at the expression inside the parentheses on both sides.
For the left-hand side (LHS):
[tex]\[ 6 + 2 \sqrt{12} \][/tex]
We can simplify [tex]\(\sqrt{12}\)[/tex] since:
[tex]\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3} \][/tex]
Thus,
[tex]\[ 6 + 2 \sqrt{12} = 6 + 2 \cdot 2 \sqrt{3} = 6 + 4 \sqrt{3} \][/tex]
Now our left-hand side expression becomes:
[tex]\[ (6 + 4 \sqrt{3})^2 \][/tex]
### Step 2: Expand the Squared Term on the Left-Hand Side
Next, we need to expand [tex]\((6 + 4 \sqrt{3})^2\)[/tex]:
[tex]\[
(6 + 4 \sqrt{3})^2 = 6^2 + 2 \cdot 6 \cdot 4 \sqrt{3} + (4 \sqrt{3})^2
\][/tex]
Calculating each term:
[tex]\[
6^2 = 36
\][/tex]
[tex]\[
2 \cdot 6 \cdot 4 \sqrt{3} = 48 \sqrt{3}
\][/tex]
[tex]\[
(4 \sqrt{3})^2 = 16 \cdot 3 = 48
\][/tex]
So,
[tex]\[
(6 + 4 \sqrt{3})^2 = 36 + 48 \sqrt{3} + 48
\][/tex]
Combine the constant terms:
[tex]\[
36 + 48 + 48 \sqrt{3} = 84 + 48 \sqrt{3}
\][/tex]
Thus, the left-hand side becomes:
[tex]\[
(6 + 2 \sqrt{12})^2 = 84 + 48 \sqrt{3}
\][/tex]
### Step 3: Simplify the Right-Hand Side
Now, let's simplify the right-hand side (RHS):
[tex]\[ 12(7 + 4 \sqrt{3}) \][/tex]
Distribute the 12:
[tex]\[
12 \cdot 7 + 12 \cdot 4 \sqrt{3} = 84 + 48 \sqrt{3}
\][/tex]
### Step 4: Equating Both Sides
We have:
[tex]\[
\text{LHS} = 84 + 48 \sqrt{3}
\][/tex]
[tex]\[
\text{RHS} = 84 + 48 \sqrt{3}
\][/tex]
Thus,
[tex]\((6 + 2 \sqrt{12})^2 = 12(7 + 4 \sqrt{3})\)[/tex].
This completes the proof.
