Exercises: Elementary operations

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

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

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

Expand the following algebraic expressions using the corresponding algebraic identities:

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

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.

In an initial assessment test in 1.º de Bachillerato, Mateo must calculate the numerical value of the following combined operation respecting the order of operations:

Mateo writes on his exam:

"$-3^2 = 9$. Then, inside the parentheses, I first calculate $5 - 3 = 2$, and $2 \cdot 2 = 4$, so it becomes $4^3 = 64$. Therefore, $A = 9 + 2 \cdot 64 - (-16) = 9 + 128 + 16 = 153$".

1. a) Identify and explain with reasoned arguments the three conceptual errors made by Mateo (one regarding the sign of powers, one regarding the order of operations inside the parentheses, and another regarding the sign when subtracting the final power).
2. b) Carry out the correct step-by-step calculation until obtaining the exact numerical value of $A$.

A youth association in Madrid has an annual budget for healthy leisure activities and decides to allocate it as follows: $\frac{3}{10}$ of the total is allocated to a chess tournament, $\frac{1}{4}$ to a drama workshop, and $\frac{2}{5}$ to a trip to the sierra de Guadarrama.

1. a) Without performing operations with a common denominator on paper yet, estimate mentally and with reasoned arguments whether these three expenses will exhaust the entire budget or if a small part will still be left free (you can help yourself by mentally associating each fraction with simple percentages: for example, $\frac{1}{4} = 25\%$).
2. b) Set up the sum of fractions, find the common denominator, and obtain the exact irreducible fraction of the budget that has been committed, as well as the remaining fraction available.
3. c) Compare the exact result with your previous estimate and justify whether your initial assessment was correct.

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, $54000$ euros remain for other operational expenses, set up the necessary calculations with fractions to determine the total budget of the company.

The side of a square plot intended for installing solar panels measures $3x - 2$ metres. If it is decided to extend the land by increasing each side by $4$ metres, 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.

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

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 $2$.

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.

You want to construct an algebraic fraction $F(x) = \frac{P(x)}{Q(x)}$ that simultaneously satisfies the following four conditions:

1. The numerator $P(x)$ is an expanded difference of squares of the form $x^2 - a^2$, where $a$ is a natural number.

2. The denominator $Q(x)$ is an expanded quadratic trinomial, resulting from squaring a binomial of the form $(x + b)^2$, where $b$ is a natural number.

3. When factoring the numerator and denominator and cancelling the appropriate common factors (never cancelling individual terms), the resulting simplified irreducible fraction is $\frac{x - 6}{x + 6}$.

4. For $x = 0$, the simplified fraction takes the value $-1$.

Deduce with reasoned arguments what the expanded polynomials $P(x)$ and $Q(x)$ are, explain the procedure to find the values of $a$ and $b$, and determine for what real value of $x$ the original fraction is not defined.

According to data from the Confederación Hidrográfica del Guadalquivir on the state of water reserves, a reservoir was at $\frac{2}{3}$ of its total capacity $C$ at the beginning of the year. During a dry spell, the volume of stored water was reduced by half. Subsequently, rainfall added a volume of water equivalent to $\left(\frac{3}{2}\right)^{-2} \cdot C$. Finally, to maintain the river's ecological flow, one-sixth of the volume contained in the reservoir after the rain was released.

1. a) Model the situation using a single combined expression of fractions and powers that determines the final volume of water in terms of $C$.
2. b) Apply the properties of negative powers and the order of operations to simplify this expression and obtain the irreducible fraction of the total capacity $C$ that remains in the reservoir.

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 millimetres. 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 $(a+b)^2 - (a-b)^2 = 4ab$.

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 $3 \cdot 2^{40}$ bytes, set up and solve the necessary operations to determine the total capacity of the server, expressing the result as a power of base $2$.

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.

To calculate the exact numerical value of the expression

two 1.º de Bachillerato students propose two different methods without using a calculator:

• Sergio's method: Calculate by hand the squares $2025^2$ and $2021^2$, subtract them, square the difference, and divide the result by the product $64 \cdot 2023^2$.

• Elena's method: Notice that $2025 = 2023 + 2$ and $2021 = 2023 - 2$, so that the bracket takes the algebraic form $[(x + y)^2 - (x - y)^2]^2$, simplify that theoretical structure using special products, and only substitute the numerical values at the end.

1. a) Algebraically expand the general theoretical expression $T = [(x + y)^2 - (x - y)^2]^2$ applying special products and factoring out common factors, and show that it reduces exactly to $16x^2 y^2$.
2. b) Use the previous result with $x = 2023$ and $y = 2$ to obtain the exact numerical value of $E$ following Elena's method.
3. c) Compare both methods and critically justify why the algebraic method is qualitatively superior to direct arithmetic calculation in terms of operational effort and error prevention.

In a telecommunications laboratory in Barcelona, the equivalent impedance $Z$ of a frequency filter is modeled using the algebraic fraction:

A student proposes the following simplification:

"Since $x^{-2}$ is the square of $x^{-1}$ and $y^{-2}$ is the square of $y^{-1}$, I directly cancel the terms in the numerator with those in the denominator by subtracting exponents, so $Z(x, y) = x^1 - y^1 = x - y$".

1. a) Explain in detail the conceptual errors in this proposal, clarifying why it is not valid to cancel individual terms in a subtraction and why the quotient of powers rules do not apply directly to terms being added or subtracted.
2. b) Transform the original expression by writing the negative powers as ordinary fractions, find the common denominator in the numerator and denominator, and simplify the complex fraction by factoring the resulting difference of squares.
3. c) Prove that the same result is reached if, instead of converting to fractions, the original denominator is factored directly as a difference of squares of powers: $x^{-2} - y^{-2} = (x^{-1} - y^{-1})(x^{-1} + y^{-1})$. Determine the restrictions that $x$ and $y$ must satisfy for the simplification to be valid.