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Let's tackle each part step-by-step.

### Question 4.1 - Simplify the following

#### 4.1.1
\[
\frac{2x^3y \times 3x^2y}{12x^6y^3}
\]

1. Multiply the numerators: \(2 \times 3 = 6\), \(x^3 \times x^2 = x^{5}\), and \(y \times y = y^2\).
2. Now we have: 
   \[
   \frac{6x^5y^2}{12x^6y^3}
   \]
3. Simplify by dividing both the coefficients and the variables:
   - \( \frac{6}{12} = \frac{1}{2} \)
   - For \( x \), \(x^5 / x^6 = x^{-1} = \frac{1}{x}\)
   - For \( y \), \(y^2 / y^3 = y^{-1} = \frac{1}{y}\)
   
So, the final simplified expression is:
\[
\frac{1}{2xy}
\]

#### 4.1.2
\[
\left( \sqrt{x^3} - 1 \right)^0
\]

Since any expression raised to the power of 0 is 1 (as long as the base is not zero), the answer is simply:
\[
1
\]

#### 4.1.3
\[
(2a + b)a - 2a(a - 3b)
\]

1. Distribute \(a\) in the first term:
   \[
   2a^2 + ab
   \]
2. Distribute \(2a\) in the second term:
   \[
   - 2a^2 + 6ab
   \]
3. Combine like terms:
   \[
   (2a^2 - 2a^2) + (ab + 6ab) = 7ab
   \]

So, the simplified answer is:
\[
7ab
\]

### Question 4.2 - Solve for \(x\)

#### 4.2.1
\[
72 = 6(x + 1)
\]

1. Divide both sides by 6:
   \[
   12 = x + 1
   \]
2. Subtract 1 from both sides:
   \[
   x = 11
   \]

#### 4.2.2
\[
\frac{2x}{3} - 1 = 15
\]

1. Add 1 to both sides:
   \[
   \frac{2x}{3} = 16
   \]
2. Multiply both sides by 3:
   \[
   2x = 48
   \]
3. Divide by 2:
   \[
   x = 24
   \]

#### 4.2.3
\[
\frac{x^2}{3} = \frac{25}{3}
\]

1. Multiply both sides by 3 to eliminate the fraction:
   \[
   x^2 = 25
   \]
2. Take the square root of both sides:
   \[
   x = \pm 5
   \]

Let me know if you would like a more detailed breakdown for any step or have questions. Here are five related questions for further practice:

1. How do you simplify expressions involving multiple variables?
2. What is the rule for any number raised to the power of zero?
3. How do you solve linear equations involving fractions?
4. What is the process for distributing terms in polynomials?
5. How do you handle equations with squares and square roots?

**Tip:** Always simplify expressions step-by-step to avoid mistakes, especially when working with exponents and fractions.

Simplify the following. Show all calculations.

1. \(\frac{2x^3y \times 3x^2y}{12x^6y^3}\)
2. \((\sqrt{x^3} - 1)^0\)
3. \((2a + b)a - 2a(a - 3b)\)

Solve for x. Show all calculations.

1. 72 = 6(x + 1)
2. \(\frac{2x}{3} - 1 = 15\)
3. \(\frac{x^2}{3} = \frac{25}{3}\)

Learn how to simplify algebraic expressions and solve equations step-by-step. This problem set covers key mathematical concepts including exponent rules, polynomial distribution, and solving linear and quadratic equations. Suitable for students in Grades 8-10.

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