Let's simplify the given expression step-by-step: [tex]\(2 \sqrt{45} + 3 \sqrt{5} + \sqrt{12} - 6 \sqrt{3}\)[/tex].
### Step 1: Simplify individual square roots
First, let's simplify each radical term individually if possible.
1. [tex]\( \sqrt{45} \)[/tex]:
[tex]\[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \][/tex]
Therefore,
[tex]\[ 2 \sqrt{45} = 2 \times 3 \sqrt{5} = 6 \sqrt{5} \][/tex]
2. [tex]\( \sqrt{12} \)[/tex]:
[tex]\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \][/tex]
Therefore,
[tex]\[ \sqrt{12} = 2 \sqrt{3} \][/tex]
### Step 2: Rewrite the expression with simplified radicals
Let's substitute back the simplified radicals into the expression:
[tex]\[ 6 \sqrt{5} + 3 \sqrt{5} + 2 \sqrt{3} - 6 \sqrt{3} \][/tex]
### Step 3: Combine like terms
Now we will combine the like terms:
- Combine the [tex]\( \sqrt{5} \)[/tex] terms:
[tex]\[ 6 \sqrt{5} + 3 \sqrt{5} = (6 + 3) \sqrt{5} = 9 \sqrt{5} \][/tex]
- Combine the [tex]\( \sqrt{3} \)[/tex] terms:
[tex]\[ 2 \sqrt{3} - 6 \sqrt{3} = (2 - 6) \sqrt{3} = -4 \sqrt{3} \][/tex]
### Step 4: Write the final simplified expression
The simplified expression is:
[tex]\[ 9 \sqrt{5} - 4 \sqrt{3} \][/tex]
So, [tex]\(2 \sqrt{45} + 3 \sqrt{5} + \sqrt{12} - 6 \sqrt{3} = 9 \sqrt{5} - 4 \sqrt{3}\)[/tex].
