Exercises: Monomials

To understand how mathematical expressions work in economics, it is essential to identify their components. Complete the requested information for the following monomials:

1. a) $7x^3$

1. b) $-x^2y$

1. c) $\frac{3}{5}ab^4$

For each of them, state the coefficient, the literal part and the degree.

In user interface design, variables are sometimes used to define the size of elements. In order to add or subtract two elements, their monomials must be like terms.

Determine which of the following pairs of monomials are like terms:

1. a) $5x^2y$ and $-2x^2y$

1. b) $3ab^2$ and $3a^2b$

1. c) $-x$ and $\frac{1}{2}x$

1. d) $4z^3$ and $4z^2$

Simplify the following expressions by collecting like terms, just as you would when calculating the total balance of income and expenses for a small shop:

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

1. b) $7ab - 3ab + ab$

1. c) $4x^3 - 5x + 2x^3 + 9x$

An architect is designing a rectangular window for a bank office. The dimensions of the window are given as a function of a variable $x$ as shown in the image. Write the monomial that represents the area of the window.

Carry out the following monomial multiplications. Remember that to multiply monomials, you multiply the coefficients and add the exponents of the same variables:

1. a) $(4x^2) \cdot (3x^5)$

1. b) $(-2a^3b) \cdot (5ab^2)$

1. c) $(\frac{1}{2}x) \cdot (6x^2y)$

Divide the following monomials and simplify the result. Keep in mind that to divide monomials, you divide the coefficients and subtract the exponents of the powers with the same base:

1. a) $\frac{12x^7}{3x^2}$

1. b) $\frac{-20a^4b^3}{5a^2b}$

1. c) $\frac{8x^3y^2}{2x^3y}$

Elena is calculating the packaging costs of a small family winery and needs to simplify the following algebraic expression:

On her exam, Elena writes:

1) "I operate the first two terms: $3x^2 y^3 + 2x^2 y^2 \cdot y = 3x^2 y^3 + 2x^2 y^3 = 5x^4 y^6$."

2) "I calculate the power of the third term: $(2x y^2)^2 = 4x^2 y^4$."

3) "I subtract the previous result: $5x^4 y^6 - 4x^2 y^4 = 1x^2 y^2$."

Identify and explain in a reasoned manner the two conceptual errors that Elena made (one regarding the addition of like monomials and another regarding the subtraction of unlike monomials). Next, carry out the correct simplification step by step yourself and indicate whether the final expression is a monomial or a binomial, specifying its degree.

A digital advertising algorithm on social networks estimates the advertising impact of a campaign using the monomial $I(u, t) = 0,4 \cdot u^2 \cdot t^3$, where $u$ is the number of users reached (in thousands) and $t$ is the average time spent on the application (in minutes).

An agency anticipates a technical update that will double the number of users ($u' = 2u$) and triple the average time spent ($t' = 3t$).

1) Estimate without calculating: before doing any numerical calculations, reason based solely on the exponents of each variable whether the total impact will be multiplied by less than 50, between 50 and 100, or by more than 100.

2) Verify formally: apply the properties of powers and the product of monomials to deduce by what exact factor $I(u, t)$ is multiplied after the changes. Compare this result with your previous estimate and explain why a variation in time $t$ has a much more drastic impact on the model than the same percentage variation in users $u$.

In a packaging design company, cardboard boxes in the shape of a rectangular prism are being designed. The length of the base is three times the width ($x$) and the height of the box is half the width.

1. Express the perimetre of the base of the box as a monomial in terms of the variable $x$.

2. Express the total volume of the box as a monomial. State the coefficient, the literal part and the degree of the monomial obtained.

Simplify the following expressions by operating with monomials and reduce the result to its simplest form:

1. a) $\displaystyle \frac{(15x^3y^2) \cdot (-2xy^4)}{6x^2y^3}$

1. b) $\displaystyle -3a^2b \cdot \left(\frac{1}{3}ab^3\right) + \frac{10a^3b^4}{2}$

A financial analyst models the profits of a tech startup using the monomial $B(x, y) = 0,45x^2y$, where $x$ represents the number of active users (in thousands) and $y$ the average marketing spend per user (in euros).

1. Calculate the numerical value of the profit if the company has $20.000$ users ($x=20$) and the marketing spend is $10$ € per user ($y=10$).

2. If the number of users doubles and the marketing spend is halved, what happens to the total profit? Justify your answer using the properties of monomials.

Determine whether the following statements are true or false, justifying your answer with examples or counterexamples:

1. a) The sum of two monomials of degree 3 is always another monomial of degree 3.
2. b) The degree of the monomial resulting from multiplying a monomial of degree $n$ by another of degree $m$ is $n + m$.
3. c) When dividing two monomials with the same literal part, the result is always a real number (degree 0).

In a rectangular plot intended for urban allotments, an automatic irrigation system is to be installed. The main plot has dimensions $5x$ long by $3y$ wide. Inside it, a square space with side length $2x$ is reserved for a tool shed that does not require watering.

1. Find the monomial that represents the total area of the plot.

2. Find the polynomial that represents the area to be covered by the irrigation system.

3. Calculate the irrigation area if $x = 4$ meters and $y = 6$ meters.

Given the monomials $M = \frac{2}{3}x^2y^3$, $N = \frac{1}{2}x^2y^3$ and $P = -\frac{5}{6}x^2y^3$:

1. Perform the operation $M - N + P$ and simplify the resulting coefficient.

2. Multiply the previous result by the monomial $Q = 12xy$. State the final degree of the expression obtained.

A team of economists is studying the efficiency of a solar power plant. The energy produced per day (in kWh) is given by the monomial $E(s, t) = \frac{3}{4}s \cdot t^2$, where $s$ is the area of the panels in square metres and $t$ is the effective sun exposure time in hours.

1. If we have a surface area $s = 8x^2$ and the exposure time is $t = 2x$, substitute these values into the formula and obtain a single monomial in terms of $x$.

2. State the degree with respect to the variable $x$ of the resulting monomial.

Find the error in the following simplification process carried out by a student and write the correct solution step by step:

Original expression: $\displaystyle \frac{8x^5 + 4x^5}{2x^2}$

Student's Step 1: $\displaystyle \frac{12x^{10}}{2x^2}$

Student's Step 2: $\displaystyle 6x^5$

Explain which law of indices or property of monomials the student applied incorrectly in each step.

In the analysis department of an e-commerce company in Madrid, the monthly gross profit (in euros) is modeled by a monomial $B(x, y)$ whose overall degree is 11. It is known that this profit is obtained by multiplying the unit margin monomial $M(x, y) = 3x^a y^2$ by the square of the sales volume monomial $V(x, y) = b x^2 y^c$, that is:

Furthermore, accounting data confirm that the final numerical coefficient of $B(x, y)$ is 48 and that the degree relative to the variable $x$ is 5.

1) Determine in a reasoned manner the values of the parameters $a$, $b$, and $c$, knowing that they are natural numbers and that $b > 0$.

2) Write the simplified analytic expression for $B(x, y)$.

3) If the company divides this profit by a fixed costs monomial $D(x, y) = 6x^3 y^2$, calculate the resulting monomial $C(x, y)$ and explain in a reasoned manner how the overall degree changes without the need to sum all the exponents from scratch again.

Two management analysts, Sergio and Rocío, have to calculate the numerical value of the projected turnover of a consulting firm. The simplified algebraic model is given by the expression:

where $x$ is the number of corporate clients and $y$ is the number of open projects. They must evaluate the result for $x = 5$ and $y = 2$.

• Sergio's method: directly substitutes the values $x = 5$ and $y = 2$ into the expression as it is and calculates the numerical powers using a calculator term by term.

• Rocío's method: first applies the properties of monomials (power, product, and quotient) to reduce the expression to a single simplified monomial $k \cdot x^n y^m$, and only then substitutes the numerical values.

1) Apply Rocío's method: simplify the expression step by step and state the resulting irreducible monomial along with its coefficient and its degree.

2) Calculate the final numerical value using the simplified expression.

3) Compare both approaches: what numerical disadvantages (risk of arithmetic overflow or calculation error) does Sergio's method present compared to Rocío's? If the consulting firm had to evaluate this formula for a table of 200 different combinations of $(x, y)$, reason which of the two methods optimizes the calculation.

In an artificial intelligence data centre, the volume of air needed to cool a server rack is modelled by the product of three critical dimensions. The height is a monomial $A(x, y) = \frac{3}{4}x^2y$, the width is $B(x, y) = \frac{8}{9}xy^3$ and the depth is $C(x, y) = \frac{1}{2}x^3y^2$.

Obtain the resulting monomial that defines the total volume $V(x, y)$ of the rack and determine its coefficient, its literal part and its overall degree. What would happen to the degree of the monomial if we doubled the exponent of the variable $x$ in the depth?

Consider the monomial $M(a, b) = k \cdot a^{n+1} \cdot b^{2m-3}$. It is known that when this monomial is divided by $Q(a, b) = \frac{1}{3}a^2b^2$, a monomial of degree 5 with coefficient 12 is obtained. Furthermore, in the resulting monomial, the degree with respect to the variable $a$ is three times the degree with respect to the variable $b$.

Calculate the values of the parameters $k$, $n$ and $m$.

Simplify the following algebraic expression until reducing it to a single monomial. Keep in mind the properties of powers and work carefully with fractional coefficients:

State the degree of the resulting monomial.

A delivery drone consumes energy depending on its speed $v$ and the mass of the payload $m$. The consumption per kilometre is given by the monomial $E(m, v) = \frac{k}{100} \cdot m \cdot v^2$.

If the drone carries a payload expressed as the monomial $m = 5x^2y$ and travels at a speed defined by the monomial $v = 2xy^3$, find the expression for the energy consumption $E$ in terms of $x$ and $y$. If the numerical value of $k$ is 0.5, calculate the consumption for $x = 2$ and $y = 1$.

Find two monomials $M_1$ and $M_2$ such that their sum is $10x^2y^3$ and their product is $16x^4y^6$. Are these monomials like terms? Justify your answer based on the definition of like terms and the results obtained.

In the design of a new circular square, the architect defines the radius of the central zone as $R = 2x^2y$. Around this zone, a landscaped ring is designed whose outer radius is three times the inner radius.

Express the area of the landscaped area (the ring) as a monomial. Leave the result in terms of $\pi$, $x$ and $y$. What is the degree of this monomial?

It is desired to distribute a government grant of $S$ euros among a number of municipalities $N$ that depends on the population. The total grant is defined by the monomial $S(a, b) = 144a^6b^4$ and the number of municipalities by $N(a, b) = (2a^2b)^2$.

Calculate the monomial that represents the amount in euros that each municipality will receive. If due to a budget adjustment, the amount per municipality must be reduced to a quarter, what would be the new resulting monomial?

Determine the values of the real coefficients $p$ and $q$ so that the product of the monomials $A(x) = p x^{n+2}$ and $B(x) = q x^{2n-1}$ is equal to the monomial $C(x) = 15x^{10}$. It is also known that if we add $A(x)$ and $B(x)$ when $n=3$, we obtain a binomial whose numerical value for $x=1$ is 8.

Solve the system to find $p$ and $q$.

A citrus export agricultural cooperative in Castellón models the weekly refrigeration energy consumption $E$ (in kWh) of its industrial cold-storage rooms through the product of factors associated with its dimensions and its operation. The room is shaped like a rectangular prism with a square base of side $x$ (in meters) and height $y$ (in meters). The weekly stock turnover is represented by the variable $z$.

The engineering technical model establishes that:

where $V(x, y)$ is the volume of the room expressed as a monomial in terms of $x$ and $y$, and $S(x, y, z)$ is a surface area and turnover index defined by the monomial $S(x, y, z) = \frac{4}{3} x y z^2$.

1) Modeling and simplification: write the volume monomial $V(x, y)$ and obtain the simplified expression for $E(x, y, z)$ as a single monomial. Specify its coefficient (as a decimal or irreducible fraction), its literal part, and its total degree.

2) Decision-making and justification: the management of the cooperative wants to expand the volumetric capacity of the refrigerated room and is considering two mutually exclusive alternatives:

• Alternative A: double only the height of the room ($y' = 2y$), keeping the base $x$ and turnover $z$ fixed.

• Alternative B: double only the side of the square base ($x' = 2x$), keeping the height $y$ and turnover $z$ fixed.

Determine algebraically, applying the properties of monomials, by what numerical factor the consumption $E$ is multiplied in each alternative. Justify which of the two expansions is more energy-efficient, that is, which one has the lower consumption per unit volume ($E/V$) after the expansion, based on the relative degrees of the variables in the resulting monomial.

In the design of an industrial electricity tariff for SMEs in the Iberian market, a consulting firm defines two cost monomials associated with energy purchases:

where $x$ represents the quote for natural gas on the MIBGAS and $y$ the price of $\text{CO}_2$ emission allowances, with $m, n \in \mathbb{N}$.

1) Determine in a reasoned manner the only values of $m$ and $n$ that make $P(x, y)$ and $Q(x, y)$ like monomials, a necessary condition for them to be added into a single cost item $C(x, y) = P(x, y) + Q(x, y)$.

2) With the values found, calculate the coefficients $a$ and $b$ knowing that the sum is $C(x, y) = 14 x^p y^q$ (stating the values of $p$ and $q$) and that the exact quotient $P(x, y) : Q(x, y)$ is equal to $2,5$.

3) An intern analyst claims: "If we multiply the resulting monomial $C(x, y)$ by any other monomial of degree $k$ and then divide the result by a monomial of degree $d$, the final expression will always be a monomial whenever $k \ge d$". Prove that this statement is false by providing a specific counterexample, and state the necessary and sufficient condition on the literal part for the division of two monomials to result in another monomial.