Elementary Operations

Carry out the following mixed operation with integers, strictly respecting the order of operations: $5 - 2 \times (4 - (-3)^2) + (-2)^3$.

Calculate and simplify fully the result of the following fractional expression: $\frac{3}{4} - \frac{1}{2} \times \left( \frac{5}{6} + \frac{1}{3} \right)$.

An agricultural irrigation water tank is filled to $\frac{3}{5}$ of its total capacity. If the equivalent of $\frac{1}{4}$ of the tank's total capacity is extracted to irrigate a field, determine what fraction of the total capacity of the tank still contains water.

A technology company allocates $\frac{2}{5}$ of its annual budget to staff salaries and $\frac{1}{3}$ of the remaining budget to marketing campaigns. If, after these two allocations, $1\,080\,000$ pesos remain for other operational expenses, set up the necessary calculations with fractions to determine the total budget of the company.

Express the result as a single power using the index laws for multiplying and dividing powers with the same base: $\frac{7^5 \times 7^{-3}}{7^4}$.

In a physical optics experiment, the intensity of a laser signal is halved each time it passes through an attenuating filter. If the original signal has an intensity $I_0$ and passes through $5$ consecutive filters, and is subsequently amplified by a factor of $\left( \frac{1}{4} \right)^{-2}$, express the final intensity of the signal in terms of $I_0$ using a single power of base $2$.

The total storage capacity of a data server is structured as follows: $\frac{1}{8}$ of the total capacity is reserved for the operating system; of the remaining capacity, a fraction equivalent to $\left( \frac{7}{4} \right)^{-1}$ is allocated to databases, and the remaining space is used for backups. If the backups occupy exactly $3 \times 2^{40}$ bytes, set up and solve the necessary operations to determine the total capacity of the server, expressing the result as a power of base $2$.

Expand the following algebraic expressions using the corresponding algebraic identities:

1. a) $(3x + 5)^2$
2. b) $(2x - 1)(2x + 1)$

The side of a square plot intended for installing solar panels measures $3x - 2$ meters. If it is decided to extend the land by increasing each side by $4$ meters, express the area of the new plot and determine the difference in area between the new plot and the old one, using the expansion of algebraic identities.

In the design of a precision mechanical part, it is specified that the area of a rectangular cross-section must be given by the formula $A = 16x^2 - 24x + 9$ square millimeters. Prove algebraically that this area always corresponds to a perfect square for any real value of $x$, and determine the expression that defines the side length of this square.

Prove rigorously, using algebraic manipulation, that the difference between the square of the sum of any two real numbers and the square of their difference is always equal to four times the product of those numbers. That is, show that $(a+b)^2 - (a-b)^2 = 4ab$.

Solve the following operation with nested fractions (compound fraction), simplifying step by step from the inside out: $\frac{1 - \frac{1}{1 - \frac{1}{2}}}{1 + \frac{1}{1 + \frac{1}{2}}}$.

Given the algebraic expression $P(x,y) = \frac{x^2 - y^2}{(x-y)^2}$, evaluate its exact value for $x = \frac{1}{2}$ and $y = -\frac{1}{3}$. Simplify the expression theoretically before substituting the values to make the numerical calculation easier.

While solving a kinematics problem, a student made the following algebraic simplification: $\frac{x^2 + y^2}{x+y} = x+y$.

Explain using an explicit numerical counterexample why this step is incorrect. Next, expand the expression $(x+y)^2$ to justify algebraically the error made, and state the single, exceptional condition on the variables $x$ and $y$ under which the student's statement would be numerically true.