Let's break down and solve the given problems one by one.

(i)

$(10x^2y^5)(-2x^4y)(-5x^{10}y^7) = Ax^B y^C$

• First, multiply the constants: $10 \times (-2) \times (-5) = 100$.
• For the powers of $x$: Add the exponents $2 + 4 + 10 = 16$.
• For the powers of $y$: Add the exponents $5 + 1 + 7 = 13$.

Thus, the equation becomes: $100x^{16}y^{13} = Ax^By^C$ Comparing both sides, we get:

• $A = 100$
• $B = 16$
• $C = 13$

(ii)

$\left( \frac{2}{3} a^3 b^6 \right) \left( \frac{5}{2} a^4 b^4 \right) \left( \frac{3}{5} a^M b^N \right) = a^{17} b^{19}$

• First, multiply the constants: $\frac{2}{3} \times \frac{5}{2} \times \frac{3}{5} = 1$.
• For $a$: Add the exponents $3 + 4 + M = 17$, giving $M = 10$.
• For $b$: Add the exponents $6 + 4 + N = 19$, giving $N = 9$.

Thus, $M = 10$ and $N = 9$.

(iii)

$(1.2m^2n^5)(0.5m^6n^3)(10m^3n^2) = Km^{11}n^{10}$

• First, multiply the constants: $1.2 \times 0.5 \times 10 = 6$.
• For $m$: Add the exponents $2 + 6 + 3 = 11$.
• For $n$: Add the exponents $5 + 3 + 2 = 10$.

Thus, $K = 6$.

(iv)

$(-11x^{10}y^{11})(-9x^{15}y^9)(2x^5y^{10})(x^6y^2) = 198A$

• First, multiply the constants: $-11 \times -9 \times 2 \times 1 = 198$.
• For $x$: Add the exponents $10 + 15 + 5 + 6 = 36$.
• For $y$: Add the exponents $11 + 9 + 10 + 2 = 32$.

Thus, the equation becomes: $198x^{36}y^{32} = 198A$ So, $A = x^{36} y^{32}$.

If you'd like more details or have any questions, feel free to ask!

Here are 5 related questions:

1. What are the laws of exponents that apply to multiplication of terms with the same base?
2. How can constants be combined when multiplying fractions?
3. Can you derive formulas to simplify expressions involving multiple terms with exponents?
4. How would the process change if some of the terms involved division instead of multiplication?
5. What happens to negative exponents during multiplication?

Tip: Always double-check the signs when multiplying or simplifying constants to avoid common mistakes!