Решение:

1) $$n^2m^5 \cdot \frac{n^2p}{m^4q} = \frac{n^2m^5n^2p}{m^4q} = \frac{n^{2+2}m^5p}{m^4q} = \frac{n^4m^5p}{m^4q} = \frac{n^4m^{5-4}p}{q} = \frac{n^4mp}{q}$$

Ответ: $$\frac{n^4mp}{q}$$

2) $$x^3y^4 \cdot \frac{xz}{y^3u} = \frac{x^3y^4xz}{y^3u} = \frac{x^{3+1}y^4z}{y^3u} = \frac{x^4y^4z}{y^3u} = \frac{x^4y^{4-3}z}{u} = \frac{x^4yz}{u}$$

Ответ: $$\frac{x^4yz}{u}$$

3) $$\frac{a^3c^2}{b^5d} \cdot a^2b^2 = \frac{a^3c^2a^2b^2}{b^5d} = \frac{a^{3+2}c^2b^2}{b^5d} = \frac{a^5c^2b^2}{b^5d} = \frac{a^5c^2}{b^{5-2}d} = \frac{a^5c^2}{b^3d}$$

Ответ: $$\frac{a^5c^2}{b^3d}$$

4) $$\frac{m^4k}{n^6p^2} \cdot m^3n^3 = \frac{m^4km^3n^3}{n^6p^2} = \frac{m^{4+3}kn^3}{n^6p^2} = \frac{m^7kn^3}{n^6p^2} = \frac{m^7k}{n^{6-3}p^2} = \frac{m^7k}{n^3p^2}$$

Ответ: $$\frac{m^7k}{n^3p^2}$$

5) $$\frac{x^7z}{y^5u} : x^3y^4 = \frac{x^7z}{y^5u} \cdot \frac{1}{x^3y^4} = \frac{x^7z}{x^3y^5uy^4} = \frac{x^7z}{x^3y^{5+4}u} = \frac{x^7z}{x^3y^9u} = \frac{x^{7-3}z}{y^9u} = \frac{x^4z}{y^9u}$$

Ответ: $$\frac{x^4z}{y^9u}$$

6) $$\frac{b^6c}{a^4d^2} : b^4a^2 = \frac{b^6c}{a^4d^2} \cdot \frac{1}{b^4a^2} = \frac{b^6c}{a^4d^2b^4a^2} = \frac{b^6c}{a^{4+2}b^4d^2} = \frac{b^6c}{a^6b^4d^2} = \frac{b^{6-4}c}{a^6d^2} = \frac{b^2c}{a^6d^2}$$

Ответ: $$\frac{b^2c}{a^6d^2}$$