Let's simplify the radical expression [tex]\(3 \sqrt{32} - 4 \sqrt{63}\)[/tex]. We will break this down into steps to make it easy to follow.
### Step 1: Simplify the individual radical terms
1. Simplify [tex]\(3 \sqrt{32}\)[/tex]:
- First, factor the number inside the square root to find its perfect square factors.
- [tex]\(32\)[/tex] can be factored as [tex]\(16 \times 2\)[/tex].
- Hence, [tex]\(\sqrt{32}\)[/tex] can be written as [tex]\(\sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2}\)[/tex].
- [tex]\(\sqrt{16}\)[/tex] is [tex]\(4\)[/tex].
- Therefore, [tex]\(\sqrt{32} = 4\sqrt{2}\)[/tex].
- Finally, multiply by 3: [tex]\(3 \sqrt{32} = 3 \times 4 \sqrt{2} = 12 \sqrt{2}\)[/tex].
2. Simplify [tex]\(4 \sqrt{63}\)[/tex]:
- Factor the number inside the square root to find its perfect square factors.
- [tex]\(63\)[/tex] can be factored as [tex]\(9 \times 7\)[/tex].
- Hence, [tex]\(\sqrt{63}\)[/tex] can be written as [tex]\(\sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7}\)[/tex].
- [tex]\(\sqrt{9}\)[/tex] is [tex]\(3\)[/tex].
- Therefore, [tex]\(\sqrt{63} = 3\sqrt{7}\)[/tex].
- Finally, multiply by 4: [tex]\(4 \sqrt{63} = 4 \times 3 \sqrt{7} = 12 \sqrt{7}\)[/tex].
### Step 2: Put it all together
Now we substitute these simplified terms back into the original expression:
[tex]\[3 \sqrt{32} - 4 \sqrt{63} = 12 \sqrt{2} - 12 \sqrt{7}\][/tex]
This is the simplified form of the original radical expression.
### Final Result
The simplified radical expression [tex]\(3 \sqrt{32} - 4 \sqrt{63}\)[/tex] is:
[tex]\[ 12 \sqrt{2} - 12 \sqrt{7} \][/tex]
Numerically, these expressions can be evaluated as:
[tex]\[ 12 \sqrt{2} \approx 16.97 \][/tex]
[tex]\[ 12 \sqrt{7} \approx 31.75 \][/tex]
Subtracting these, we get:
[tex]\[ 12 \sqrt{2} - 12 \sqrt{7} \approx -14.78 \][/tex]
Hence, the simplified radical expression is [tex]\( 12 \sqrt{2} - 12 \sqrt{7} \)[/tex], which approximates to [tex]\(-14.78\)[/tex].
