"Basic Operations: Fractions, Exponents, and Special Products"

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

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

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

An agricultural irrigation water tank is full to $\frac{3}{5}$ of its capacity. If the equivalent of $\frac{1}{4}$ of the tank's total capacity is extracted to water a plot, determine what fraction of the tank's total capacity 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 it still has $54000$ dollars left for other operating expenses, set up the necessary operations with fractions to determine the total budget of the company.

Express the result as a single power using the properties of multiplication and division of powers with the same base: $\frac{7^5 \cdot 7^{-3}}{7^4}$.

Calculate the exact value of the following numerical expression, applying the definition of a negative exponent: $\left( \frac{2}{3} \right)^{-2} + \left( \frac{1}{4} \right)^{-1}$.

Simplify the following algebraic expression by applying the properties of powers and express the final result using only positive exponents: $(x^3 \cdot x^{-5})^2$.

Calculate and fully simplify the following expression, paying special attention to negative exponents and the order of operations: $\left( \frac{3}{2} - \frac{1}{4} \right)^{-2} \cdot \left( \frac{5}{3} \right)^{-1}$.

Simplify the following algebraic expression, grouping equal bases and leaving the final result expressed exclusively with positive exponents: $\frac{(2a^{-2} b^3)^{-3}}{4a^4 b^{-5}}$.

Fully simplify the following complex algebraic expression, indicating the order of the properties of powers used and leaving the final result without negative exponents: $\left[ \frac{(x^2 y^{-1})^{-2} \cdot (x^{-3} y^2)^3}{(x y)^{-4}} \right]^{-1}$.

Simplify the following numerical fraction by first factorizing its terms into powers of prime numbers: $\frac{2^4 \cdot 3^2 \cdot 5}{2^2 \cdot 3^3 \cdot 5^2}$.

Simplify the following numerical expression by first factorizing the composite bases into prime numbers and applying the properties of powers: $\frac{18^4 \cdot 12^{-3}}{24^2 \cdot 9^{-2}}$.

Determine the value of the integer $n$ so that the following exponential equality is true, transforming all composite bases into powers of base $2$: $\frac{4^n \cdot 8^3 \cdot 2^{-5}}{16^2} = 2^{10}$.

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 multiplying 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 with 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 $\left( \frac{3}{2} \right)^{-1}$ is allocated to databases, and the leftover space is used for backups. If the backups occupy exactly $3 \cdot 2^{40}$ bytes, set up and solve the necessary operations to determine the total capacity of the server, expressing the result as a power of base $2$.

Calculate the exact value of the following numerical expression, detailing each logical step and using a combination of the properties of fractions, negative exponents and algebraic identities: $\frac{\left( 1 - \frac{3}{5} \right)^{-2} + \left( \frac{5}{2} - 1 \right)^2}{\left( \frac{2}{3} \right)^{-3} \cdot \left( \frac{1}{2} \right)^2}$.

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 the installation of solar panels measures $3x - 2$ feet. If it is decided to expand the land by increasing each side by $4$ feet, express the area of the new land and determine the difference in area between the new and old plots, 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 correspond to the formula $A = 16x^2 - 24x + 9$ square inches. 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, by 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 these numbers. That is, prove that $(a+b)^2 - (a-b)^2 = 4ab$.

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 facilitate the numerical calculation.

Subtract the following algebraic fractions and simplify the result to obtain a single irreducible fraction. Apply factorization using algebraic identities in the denominators to find the lowest common multiple: $\frac{x^2 - 1}{x^2 + 2x + 1} - \frac{x - 1}{x + 1}$.

During the resolution of a kinematics problem, a student has made the following algebraic simplification: $\frac{x^2 + y^2}{x+y} = x+y$.

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