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Let's solve the problems one by one from the provided image.

### 36. \(\frac{4}{5} \div \frac{3}{10}\)
To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
\[
\frac{4}{5} \div \frac{3}{10} = \frac{4}{5} \times \frac{10}{3}
\]
Multiply the numerators and denominators:
\[
= \frac{4 \times 10}{5 \times 3} = \frac{40}{15} = \frac{8}{3}
\]
So, the answer is \(\frac{8}{3}\).

### 37. \(\frac{12a^2}{18b^2} \div \frac{4a^2}{3b}\)
Dividing two fractions means multiplying the first fraction by the reciprocal of the second:
\[
\frac{12a^2}{18b^2} \times \frac{3b}{4a^2}
\]
Simplify the terms:
\[
= \frac{12 \times 3b}{18 \times 4a^2 \times b^2} = \frac{36b}{72a^2b^2}
\]
\[
= \frac{1}{2ab}
\]
So, the answer is \(\frac{1}{2ab}\).

### 38. \(\frac{6c^7}{(4c)} \div \frac{90c^5}{16c^6}\)
First, simplify the expressions. The first fraction becomes:
\[
\frac{6c^7}{4c} = \frac{6}{4} \times c^{7-1} = \frac{3}{2}c^6
\]
Now, the second fraction simplifies:
\[
\frac{90c^5}{16c^6} = \frac{90}{16} \times c^{5-6} = \frac{45}{8c}
\]
Now divide the two fractions:
\[
\frac{\frac{3}{2}c^6}{\frac{45}{8c}} = \frac{3}{2}c^6 \times \frac{8c}{45}
\]
\[
= \frac{3 \times 8c}{2 \times 45}c^6 = \frac{24c}{90}c^6 = \frac{4c^7}{15}
\]
So, the answer is \(\frac{4c^7}{15}\).

### 39. \(\frac{20d^2}{4} \div 5d\)
First simplify the expression:
\[
\frac{20d^2}{4} = 5d^2
\]
Now divide by \(5d\):
\[
\frac{5d^2}{5d} = d
\]
So, the answer is \(d\).

### 40. \(\frac{15e}{f^2} \div \frac{30e}{f^4}\)
Dividing these fractions, we multiply by the reciprocal of the second fraction:
\[
\frac{15e}{f^2} \times \frac{f^4}{30e}
\]
Cancel \(e\) and simplify the fractions:
\[
= \frac{15f^4}{30f^2} = \frac{f^2}{2}
\]
So, the answer is \(\frac{f^2}{2}\).

### 41. \(\frac{4g^2}{6h^2} \div \frac{8g}{12gh}\)
Multiply by the reciprocal of the second fraction:
\[
\frac{4g^2}{6h^2} \times \frac{12gh}{8g}
\]
Simplify:
\[
= \frac{4 \times 12g^3}{6 \times 8gh^2} = \frac{48g^3}{48gh^2}
\]
Cancel common terms:
\[
= \frac{g^2}{h}
\]
So, the answer is \(\frac{g^2}{h}\).

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Do you need any further details on these problems?

Here are some related questions for further practice:
1. Solve \(\frac{5}{6} \div \frac{2}{9}\).
2. Simplify \(\frac{10a^3}{15b} \div \frac{5a^2}{9b^2}\).
3. Solve \(\frac{24c^2}{8c} \div \frac{30c^4}{20c}\).
4. Simplify \(\frac{40d^3}{5d^2} \div \frac{20d}{4}\).
5. Solve \(\frac{10x}{2y} \div \frac{15x^2}{5y^3}\).

**Tip**: Always simplify the individual fractions before performing division; this will make the calculations much easier!

Perform the indicated operations: 
36. 4/5 ÷ 3/10 
37. 12a²/18b² ÷ 4a²/3b 
38. 6c⁷/(4c) ÷ 90c⁵/16c⁶ 
39. 20d²/4 ÷ 5d 
40. 15e/f² ÷ 30e/f⁴ 
41. 4g²/6h² ÷ 8g/12gh

Solve complex fraction and algebraic expressions with step-by-step instructions. Learn how to divide fractions, simplify algebraic terms, and apply exponent laws in problems such as 4/5 ÷ 3/10 and 12a²/18b² ÷ 4a²/3b. Suitable for Grades 8-10.

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