What is the following product?

[tex]\[ 3 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3}) \][/tex]

A. \( 30 \sqrt{2} - 21 \sqrt{5} \)

B. \( 60 \sqrt{2} - 21 \sqrt{5} \)

C. \( 30 \sqrt{3} - 21 \sqrt{6} \)

D. [tex]\( 60 \sqrt{3} - 21 \sqrt{6} \)[/tex]



Answer :

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]