Simplify: (2x³ - y(5x + 12y))/(5xy² + 12y³ - 2x²y)

Here are the solutions to the problems:

4. Simplify: $\frac{2x^3 - y(5x+12y)}{5xy^2 + 12y^3 - 2x^2y}$

Assumption: Based on the typical structure of such simplification problems, it is highly probable that the numerator was intended to be $2x^3 - 5x^2y - 12xy^2$ instead of $2x^3 - y(5x+12y)$. We will proceed with this assumption to allow for simplification.

Step 1: Expand the numerator and factor out common terms.
Assuming the numerator is $2x^3 - 5x^2y - 12xy^2$:

$$2x^3 - 5x^2y - 12xy^2 = x(2x^2 - 5xy - 12y^2)$$

Step 2: Factor the quadratic expression in the numerator.
We need two numbers that multiply to $2 \times (-12) = -24$ and add to $-5$. These numbers are $-8$ and 3.

$$2x^2 - 5xy - 12y^2 = 2x^2 - 8xy + 3xy - 12y^2$$ $$= 2x(x - 4y) + 3y(x - 4y)$$ $$= (2x + 3y)(x - 4y)$$

So, the numerator is $x(2x + 3y)(x - 4y)$.

Step 3: Factor the denominator.

$$5xy^2 + 12y^3 - 2x^2y$$

Factor out $y$:

$$y(5xy + 12y^2 - 2x^2)$$

Rearrange the terms inside the parenthesis:

$$y(-2x^2 + 5xy + 12y^2)$$

Step 4: Factor the quadratic expression in the denominator.
We need two numbers that multiply to $(-2) \times 12 = -24$ and add to 5. These numbers are 8 and $-3$.

$$-2x^2 + 5xy + 12y^2 = -2x^2 + 8xy - 3xy + 12y^2$$ $$= -2x(x - 4y) - 3y(x - 4y)$$ $$= (-2x - 3y)(x - 4y)$$ $$= -(2x + 3y)(x - 4y)$$

So, the denominator is $y(-(2x + 3y)(x - 4y)) = -y(2x + 3y)(x - 4y)$.

Step 5: Substitute the factored expressions back into the fraction and simplify.

$$\frac{x(2x + 3y)(x - 4y)}{-y(2x + 3y)(x - 4y)}$$

Assuming $2x+3y \neq 0$ and $x-4y \neq 0$, we can cancel the common factors:

$$-\frac{x}{y}$$

The simplified expression is $\boxed{-\frac{x}{y}}$.

5. The masses of two similar solids are 324g and 768g. Find:

First, determine the ratio of the masses. For similar solids, the ratio of their volumes (and thus masses) is equal to the cube of the ratio of their corresponding linear dimensions.
Let $M_1 = 324 \text{ g}$ (smaller solid) and $M_2 = 768 \text{ g}$ (bigger solid).
Let $L_1$ and $L_2$ be corresponding linear dimensions.

$$\frac{M_1}{M_2} = \left(\frac{L_1}{L_2}\right)^3$$ $$\frac{324}{768} = \left(\frac{L_1}{L_2}\right)^3$$

Simplify the mass ratio:

$$\frac{324 \div 12}{768 \div 12} = \frac{27}{64}$$

So,

$$\left(\frac{L_1}{L_2}\right)^3 = \frac{27}{64}$$

Take the cube root of both sides to find the linear scale factor:

$$\frac{L_1}{L_2} = \sqrt[3]{\frac{27}{64}} = \frac{3}{4}$$