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The image contains four mathematical expressions that need to be simplified without using a calculator. I'll solve each of them step by step.

### (i) \(\frac{5}{6} \div \frac{15}{8}\)
To divide fractions, we multiply by the reciprocal:
\[
\frac{5}{6} \div \frac{15}{8} = \frac{5}{6} \times \frac{8}{15} = \frac{5 \times 8}{6 \times 15} = \frac{40}{90}
\]
Now simplify \(\frac{40}{90}\):
\[
\frac{40}{90} = \frac{4}{9}
\]

Thus, \(\frac{5}{6} \div \frac{15}{8} = \frac{4}{9}\).

### (ii) \(6\sqrt{2} - 2\sqrt{8} + 5\sqrt{18}\)
First, simplify each square root term:
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}, \quad \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\]
Now substitute these back into the expression:
\[
6\sqrt{2} - 2(2\sqrt{2}) + 5(3\sqrt{2}) = 6\sqrt{2} - 4\sqrt{2} + 15\sqrt{2}
\]
Now combine like terms:
\[
(6 - 4 + 15)\sqrt{2} = 17\sqrt{2}
\]

Thus, \(6\sqrt{2} - 2\sqrt{8} + 5\sqrt{18} = 17\sqrt{2}\).

### (iii) \(\frac{6\sqrt{12}}{4\sqrt{3}}\)
Simplify the square roots:
\[
\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
\]
Substitute this back into the expression:
\[
\frac{6(2\sqrt{3})}{4\sqrt{3}} = \frac{12\sqrt{3}}{4\sqrt{3}}
\]
The \(\sqrt{3}\) terms cancel out:
\[
\frac{12}{4} = 3
\]

Thus, \(\frac{6\sqrt{12}}{4\sqrt{3}} = 3\).

### (iv) \(\sqrt{3}(\sqrt{3} - \sqrt{2}) - \sqrt{1}\)
First, simplify \(\sqrt{1} = 1\):
\[
\sqrt{3}(\sqrt{3} - \sqrt{2}) - 1
\]
Now distribute \(\sqrt{3}\):
\[
(\sqrt{3} \times \sqrt{3}) - (\sqrt{3} \times \sqrt{2}) - 1 = 3 - \sqrt{6} - 1
\]
Now simplify the constants:
\[
3 - 1 - \sqrt{6} = 2 - \sqrt{6}
\]

Thus, \(\sqrt{3}(\sqrt{3} - \sqrt{2}) - \sqrt{1} = 2 - \sqrt{6}\).

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Would you like more details on any of these steps?

### Relative Questions:
1. How do you simplify expressions involving division of fractions?
2. What is the method to simplify square roots of products like \(\sqrt{8}\) or \(\sqrt{18}\)?
3. Can you provide examples of simplifying radicals in a complex expression?
4. What is the reasoning behind multiplying by the reciprocal when dividing fractions?
5. What techniques can you use to combine terms with square roots?

### Tip:
When simplifying square roots, look for perfect square factors to simplify the expression effectively.

Simplify the following without using a calculator:
(i) \(\frac{5}{6} \div \frac{15}{8}\)
(ii) \(6\sqrt{2} - 2\sqrt{8} + 5\sqrt{18}\)
(iii) \(\frac{6\sqrt{12}}{4\sqrt{3}}\)
(iv) \(\sqrt{3}(\sqrt{3} - \sqrt{2}) - \sqrt{1}\)

This page provides a detailed step-by-step guide to simplifying mathematical expressions involving fractions and radicals without using a calculator. The problem includes fraction division, simplifying radicals, and applying the distributive property. Suitable for students in Grades 9-10.

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