Sure! Let's break down each part of the problem step by step:
### Problem (i): [tex]\( 3 \sqrt{50} \times \sqrt{192} \)[/tex]
1. Simplify the square roots:
- [tex]\(\sqrt{50}\)[/tex] can be simplified to [tex]\(5\sqrt{2}\)[/tex], because [tex]\(50 = 25 \times 2\)[/tex] and [tex]\(\sqrt{25} = 5\)[/tex].
- [tex]\(\sqrt{192}\)[/tex] can be simplified to [tex]\(8\sqrt{3}\)[/tex], because [tex]\(192 = 64 \times 3\)[/tex] and [tex]\(\sqrt{64} = 8\)[/tex].
2. Substitute the simplified forms into the expression:
[tex]\[
3 \sqrt{50} \times \sqrt{192} = 3 (5\sqrt{2}) \times (8\sqrt{3})
\][/tex]
3. Multiply the coefficients and the square roots separately:
[tex]\[
3 \times 5 \times 8 \times \sqrt{2} \times \sqrt{3} = 120 \sqrt{6}
\][/tex]
Therefore, the simplified form of [tex]\( 3 \sqrt{50} \times \sqrt{192} \)[/tex] is:
[tex]\[
\boxed{120 \sqrt{6}}
\][/tex]
When we calculate this using a numerical approach, it equals approximately [tex]\(293.93876913398134\)[/tex].
### Problem (ii): [tex]\( \frac{3}{4} \sqrt{112} \times \frac{8}{3} \sqrt{45} \)[/tex]
1. Simplify the square roots:
- [tex]\(\sqrt{112}\)[/tex] can be simplified to [tex]\(4\sqrt{7}\)[/tex], because [tex]\(112 = 16 \times 7\)[/tex] and [tex]\(\sqrt{16} = 4\)[/tex].
- [tex]\(\sqrt{45}\)[/tex] can be simplified to [tex]\(3\sqrt{5}\)[/tex], because [tex]\(45 = 9 \times 5\)[/tex] and [tex]\(\sqrt{9} = 3\)[/tex].
2. Substitute the simplified forms into the expression:
[tex]\[
\frac{3}{4} \sqrt{112} \times \frac{8}{3} \sqrt{45} = \frac{3}{4} (4 \sqrt{7}) \times \frac{8}{3} (3 \sqrt{5})
\][/tex]
3. Multiply the fractions and the square roots separately:
- The fractions part: [tex]\( \frac{3}{4} \times \frac{8}{3} = 2 \)[/tex]
- Combining the square roots: [tex]\( 4 \sqrt{7} \times 3 \sqrt{5} = 12 \sqrt{35} \)[/tex]
4. Combine both parts:
[tex]\[
\left(\frac{3}{4} \times \frac{8}{3}\right) \times \left(4 \sqrt{7} \times 3 \sqrt{5}\right) = 2 \times 12 \sqrt{35} = 24 \sqrt{35}
\][/tex]
Therefore, the simplified form of [tex]\( \frac{3}{4} \sqrt{112} \times \frac{8}{3} \sqrt{45} \)[/tex] is:
[tex]\[
\boxed{24 \sqrt{35}}
\][/tex]
When we calculate this using a numerical approach, it equals approximately [tex]\(141.9859147943908\)[/tex].
