Let's break down the problem step by step to determine the product of the given expression \(3 \sqrt{2} (5 \sqrt{6} - 7 \sqrt{3})\).
### Step 1: Distribute \(3 \sqrt{2}\)
We need to distribute \(3 \sqrt{2}\) to both terms inside the parentheses:
[tex]\[ 3 \sqrt{2} \times 5 \sqrt{6} - 3 \sqrt{2} \times 7 \sqrt{3} \][/tex]
### Step 2: Simplify the First Term
Consider the product \(3 \sqrt{2} \times 5 \sqrt{6}\):
[tex]\[ 3 \times 5 \times \sqrt{2} \times \sqrt{6} = 15 \times \sqrt{12} \][/tex]
We know that \(\sqrt{12}\) can be simplified further:
[tex]\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \][/tex]
Thus,
[tex]\[ 15 \times \sqrt{12} = 15 \times 2 \sqrt{3} = 30 \sqrt{3} \][/tex]
### Step 3: Simplify the Second Term
Consider the product \(3 \sqrt{2} \times 7 \sqrt{3}\):
[tex]\[ 3 \times 7 \times \sqrt{2} \times \sqrt{3} = 21 \times \sqrt{6} \][/tex]
### Step 4: Combine Both Terms
Now let's combine the simplified terms:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
### Conclusion
The given product \(3 \sqrt{2} (5 \sqrt{6} - 7 \sqrt{3})\) is equal to \(30 \sqrt{3} - 21 \sqrt{6}\).
Therefore, the correct answer from the provided options is:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
