Answer :
To find the solution to the expression [tex]\( 3 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3}) \)[/tex], we can use the distributive property of multiplication over subtraction. Let's break it down step-by-step:
### Step 1: Distribute the term [tex]\( 3 \sqrt{2} \)[/tex] to each term inside the parenthesis:
1. Distribute [tex]\( 3 \sqrt{2} \)[/tex] to [tex]\( 5 \sqrt{6} \)[/tex]:
[tex]\[ 3 \sqrt{2} \cdot 5 \sqrt{6} \][/tex]
2. Distribute [tex]\( 3 \sqrt{2} \)[/tex] to [tex]\( -7 \sqrt{3} \)[/tex]:
[tex]\[ 3 \sqrt{2} \cdot -7 \sqrt{3} \][/tex]
### Step 2: Calculate each resulting product:
1. For [tex]\( 3 \sqrt{2} \cdot 5 \sqrt{6} \)[/tex]:
[tex]\[ 3 \cdot 5 \cdot \sqrt{2 \cdot 6} = 15 \cdot \sqrt{12} \][/tex]
Simplify [tex]\( \sqrt{12} \)[/tex]:
[tex]\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3} \][/tex]
So,
[tex]\[ 15 \cdot \sqrt{12} = 15 \cdot 2 \sqrt{3} = 30 \sqrt{3} \][/tex]
2. For [tex]\( 3 \sqrt{2} \cdot -7 \sqrt{3} \)[/tex]:
[tex]\[ 3 \cdot -7 \cdot \sqrt{2 \cdot 3} = -21 \cdot \sqrt{6} \][/tex]
### Step 3: Combine the results:
Combining both resulting terms from the distribution, we have:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
### Conclusion:
The product [tex]\( 3 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3}) \)[/tex] simplifies to [tex]\( 30 \sqrt{3} - 21 \sqrt{6} \)[/tex]. Thus, the correct answer from the given options is:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
So, the solution is [tex]\( 30 \sqrt{3} - 21 \sqrt{6} \)[/tex].
### Step 1: Distribute the term [tex]\( 3 \sqrt{2} \)[/tex] to each term inside the parenthesis:
1. Distribute [tex]\( 3 \sqrt{2} \)[/tex] to [tex]\( 5 \sqrt{6} \)[/tex]:
[tex]\[ 3 \sqrt{2} \cdot 5 \sqrt{6} \][/tex]
2. Distribute [tex]\( 3 \sqrt{2} \)[/tex] to [tex]\( -7 \sqrt{3} \)[/tex]:
[tex]\[ 3 \sqrt{2} \cdot -7 \sqrt{3} \][/tex]
### Step 2: Calculate each resulting product:
1. For [tex]\( 3 \sqrt{2} \cdot 5 \sqrt{6} \)[/tex]:
[tex]\[ 3 \cdot 5 \cdot \sqrt{2 \cdot 6} = 15 \cdot \sqrt{12} \][/tex]
Simplify [tex]\( \sqrt{12} \)[/tex]:
[tex]\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2 \sqrt{3} \][/tex]
So,
[tex]\[ 15 \cdot \sqrt{12} = 15 \cdot 2 \sqrt{3} = 30 \sqrt{3} \][/tex]
2. For [tex]\( 3 \sqrt{2} \cdot -7 \sqrt{3} \)[/tex]:
[tex]\[ 3 \cdot -7 \cdot \sqrt{2 \cdot 3} = -21 \cdot \sqrt{6} \][/tex]
### Step 3: Combine the results:
Combining both resulting terms from the distribution, we have:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
### Conclusion:
The product [tex]\( 3 \sqrt{2}(5 \sqrt{6} - 7 \sqrt{3}) \)[/tex] simplifies to [tex]\( 30 \sqrt{3} - 21 \sqrt{6} \)[/tex]. Thus, the correct answer from the given options is:
[tex]\[ 30 \sqrt{3} - 21 \sqrt{6} \][/tex]
So, the solution is [tex]\( 30 \sqrt{3} - 21 \sqrt{6} \)[/tex].
