Let's go through the questions in the image and solve them step by step:

1. Work out the value of each expression when $x = 3$, $y = 5$, and $z = -2$.

a) $x(3y + z)$

$x(3y + z) = 3(3 \times 5 + (-2)) = 3(15 - 2) = 3 \times 13 = 39$

b) $(2x)^2 + y^3$

$(2x)^2 + y^3 = (2 \times 3)^2 + 5^3 = 6^2 + 125 = 36 + 125 = 161$

c) $\frac{(x + y)}{4} - 5z$

$\frac{(x + y)}{4} - 5z = \frac{(3 + 5)}{4} - 5(-2) = \frac{8}{4} + 10 = 2 + 10 = 12$

2. Write an expression for the perimeter and area of this rectangle.

The rectangle dimensions are $3x + 4$ and $5x$.

• Perimeter: $2 \times (\text{length} + \text{width})$ $2 \times ((3x + 4) + 5x) = 2 \times (8x + 4) = 16x + 8$
• Area: $\text{length} \times \text{width}$ $(3x + 4) \times 5x = 15x^2 + 20x$

3. Simplify each expression.

a) $x^2 \times x^3$

$x^2 \times x^3 = x^{2+3} = x^5$

b) $q^8 - q^2$

This cannot be simplified further unless factored: $q^8 - q^2 = q^2(q^6 - 1)$

c) $(h^2)^5$

$(h^2)^5 = h^{2 \times 5} = h^{10}$

d) $3m^7 \times 5m^2$

$3m^7 \times 5m^2 = 15m^{7+2} = 15m^9$

e) $\frac{12u^5}{6u^3}$

$\frac{12u^5}{6u^3} = 2u^{5-3} = 2u^2$

f) $4p^2 - p^2$

$4p^2 - p^2 = 3p^2$

4. Expand and simplify each expression.

a) $(x + 2)(x + 5)$

$(x + 2)(x + 5) = x^2 + 5x + 2x + 10 = x^2 + 7x + 10$

b) $(x - 3)(x + 4)$

$(x - 3)(x + 4) = x^2 + 4x - 3x - 12 = x^2 + x - 12$

c) $(x + 6)(x - 9)$

$(x + 6)(x - 9) = x^2 - 9x + 6x - 54 = x^2 - 3x - 54$

d) $(x - 10)(x - 4)$

$(x - 10)(x - 4) = x^2 - 4x - 10x + 40 = x^2 - 14x + 40$

e) $(x - 8)(x + 8)$

This is a difference of squares: $(x - 8)(x + 8) = x^2 - 64$

f) $(x + 6)(x - 9)$

(Same as part c): $x^2 - 3x - 54$

5. Simplify each expression.

a) $\frac{x}{3} + \frac{x}{3}$

$\frac{x}{3} + \frac{x}{3} = \frac{2x}{3}$

b) $\frac{y}{5} - \frac{y}{15}$

$\frac{y}{5} - \frac{y}{15} = \frac{3y}{15} - \frac{y}{15} = \frac{2y}{15}$

c) $\frac{3x}{5} - \frac{y}{20}$

No like terms, so it cannot be simplified further.

d) $\frac{9x - 15}{3}$

$\frac{9x - 15}{3} = 3x - 5$

6.

a) Use the formula $x = y^2 + 5z$ to work out the value of $x$ when $y = 4$ and $z = 3$.

$x = 4^2 + 5(3) = 16 + 15 = 31$

b) Make $z$ the subject of the formula $x = y^2 + 5z$.

$x = y^2 + 5z \implies z = \frac{x - y^2}{5}$ When $x = 55$ and $y = 5$: $z = \frac{55 - 5^2}{5} = \frac{55 - 25}{5} = \frac{30}{5} = 6$

c) Make $y$ the subject of the formula and work out the value of $y$ when $x = 46$ and $z = 2$.

Formula: $x = y^2 + 5z \implies y = \sqrt{x - 5z}$ When $x = 46$ and $z = 2$: $y = \sqrt{46 - 5(2)} = \sqrt{46 - 10} = \sqrt{36} = 6$

Would you like more detailed explanations or have any questions?

Related questions:

1. How do you factor quadratic expressions efficiently?
2. What are the steps for simplifying radical expressions?
3. Can you explain more about the difference of squares and how it's used in algebra?
4. How do you determine the subject of a formula in different cases?
5. Could you go over more examples of perimeter and area problems?

Tip:

When simplifying algebraic expressions, always look for common factors or terms that can be combined to simplify your calculations.