Exercises: Monomials

Identify the coefficient, the variable part and the degree of the monomial $-\frac{3}{5}x^2y^3z$.

Perform the following addition and subtraction of like monomials and simplify the result:

$7a^2b - 3a^2b + \frac{1}{2}a^2b$

Multiply the following monomials and express the result in the most simplified form possible:

$(-4x^3y) \cdot (5xy^4)$

Divide the following monomials, clearly indicating the coefficient and the variable part of the result:

$\frac{18p^5q^3}{3p^2q}$

Calculate the power of the following monomial by applying index laws:

$(-2m^3n^4)^5$

Find the numerical value of the monomial $M(x,y) = 4x^2y^3$ for the values $x = -2$ and $y = 3$.

Write the monomial that mathematically represents the following statement: "Half the product of the square of a number and the cube of a different number".

Check whether the monomials $A = 5x^2y$ and $B = -2yx^2$ are like terms. If so, calculate their sum $A + B$ and their difference $A - B$.

[SOMEONE ELSE'S ERROR] Lucas is reviewing for his Mathematics test and writes the following statement in his notebook:

"The monomials $A = -4x^3y^2$ and $B = 7x^2y^3$ are similar because both are formed by the letters $x$ and $y$ and both have degree $5$. Therefore, we can add them directly, obtaining $3x^5y^5$".

1. a) Locate and explain in detail the two conceptual errors that Lucas made (one concerning the definition of similar monomials and another concerning the procedure for adding monomials).
2. b) Write a monomial $C$ that is indeed similar to $A$, whose coefficient is an irreducible negative fraction, and find its numerical value for $x = -1$ and $y = 2$.

[MODEL] According to road safety data from the Dirección General de Tráfico (DGT), the kinetic energy of a moving vehicle is modeled using the physics formula $E_c = \frac{1}{2}m \cdot v^2$, where $m$ represents the vehicle's mass in kilograms ($\text{kg}$) and $v$ the speed at which it travels in meters per second ($\text{m/s}$).

A goods delivery van in Madrid has a fixed mass of $m = 2400\text{ kg}$.

1. a) Write the expression for the van's kinetic energy as a monomial in terms of the variable $v$. Clearly identify its coefficient, its literal part, and its degree.
2. b) On a residential street with a speed limit of $30\text{ km/h}$, the van travels at about $v = 8\text{ m/s}$. What would happen to the value of the kinetic energy if the driver doubled their speed to $16\text{ m/s}$? Justify your answer based on the degree of the variable $v$, without having to perform the entire calculation again.

A swimming pool is in the shape of a rectangular prism with the dimensions shown in the figure. Express the volume of the pool as a monomial. Then, calculate the amount of water (in litres) it contains when completely full if $x = 3$ metres. (Remember that $1 \text{ m}^3 = 1000 \text{ litres}$).

The kinetic energy of a moving object is calculated using the expression $E_c = \frac{1}{2}mv^2$. If the mass of a particle is given by the monomial $m = 4t^2$ kg and its speed is $v = 3t^3$ m/s, express its kinetic energy as a simplified monomial in terms of time $t$.

A technology company determines that the cost of manufacturing an electronic component is given by the monomial $C(x) = 1,5x^3$ euros, where $x$ is the number of batches produced. If the revenue obtained from selling those batches is expressed by the monomial $I(x) = 4,2x^3$ euros, write the monomial representing the profit $B(x)$ and calculate this profit if $x = 10$ batches are produced.

Simplify the following combined expression of monomials, first calculating the product in the numerator and then performing the division:

$\frac{(3a^2b^3) \cdot (-4a^4b)}{6a^3b^2}$

Perform the following combined operation respecting the order of operations and collect like terms:

$(2x^2y)^3 - 5x^6y^3 + (-x^2y)^3$

Find the missing monomial $M(x)$ so that the following equality is true, justifying the steps taken:

$M(x) \cdot (7x^4) = -35x^9$

The degree of the monomial $P(x,y) = -8x^{2m}y^{m+1}$ is $13$. Set up a linear equation, find the value of the parameter $m$ and write the final expression of the monomial.

A photo frame is shaped like an annulus. If the outer radius is $R = 5x$ and the inner radius is $r = 3x$, express the total area of the frame as a monomial in terms of $x$. (Leave the result in terms of the number $\pi$).

[INVERSE PROBLEM] Find two monomials, which we will call $P$ and $Q$, that satisfy all and each of the following conditions simultaneously:

1. The literal part of both monomials consists exclusively of the variables $a$, $b$, and $c$.

2. Monomial $P$ has an overall degree of $7$. In it, the exponent of $a$ is twice that of $b$, and the exponent of $c$ is equal to $1$.

3. The coefficient of $P$ is the negative integer $-3$.

4. Monomial $Q$ is similar to $P$, and its coefficient is a positive integer $k$.

5. The numerical value of the sum monomial $(P + Q)$ when substituting $a = 1$, $b = -1$, and $c = 2$ is exactly equal to $8$.

Show step by step the reasoning followed to deduce the exponents and calculate the value of the coefficient $k$ of $Q$. Furthermore, justify whether it would be possible to find an integer value for $k$ if the required numerical result had been $7$ instead of $8$.

[MODEL + ESTIMATE AND CHECK] In an irrigation community in the Guadalquivir valley, an automated drip irrigation system is installed to optimize water usage during a drought period. The monthly volume of water consumed $V$ (in cubic meters, $\text{m}^3$) is modeled by the monomial:

where $n$ is the number of open irrigation sectors, $h$ is the daily operating hours, and $d$ is the internal diameter of the main pipes (measured in decimeters, $\text{dm}$).

1. a) What is the total degree of the monomial? Explain what real geometric meaning the variable $d$ having an exponent of two has in the formula.
2. b) Estimate before calculating: If the community decides to upgrade the system and replace the pipes of $d = 1\text{ dm}$ with ones of $d = 2\text{ dm}$, keeping the sectors ($n$) and the hours ($h$) the same, a farmer claims that "water usage will simply double". Make a reasoned qualitative estimate: is the farmer right, or will the increase be greater? Justify it based on the structure of the monomial.
3. c) Check: Calculate the exact numerical value of $V$ in both cases for a farm with $n = 5$ sectors and $h = 4$ daily hours. Determine by what factor the water consumption is actually multiplied when the diameter is doubled.

A sphere of radius $r = x^2$ is inserted into a cylinder with base radius $R = x^2$ and height $h = 2x^2$, such that the sphere fits perfectly. Express the volume of the empty space remaining between the cylinder and the sphere as a simplified monomial in terms of $x$ and $\pi$.

Given the monomials $A = k \cdot x^a y^5$ and $B = 4x^2 y^b$, it is known that their product is the monomial $M = -12x^7 y^8$. Determine the numerical values of the parameters $k$, $a$ and $b$. Once found, calculate the numerical value of the monomial resulting from the division $\frac{A}{B}$ for $x = -1$ and $y = 2$.

The gravitational force of attraction between two bodies is given by $F = G \frac{m_1 m_2}{d^2}$, where the universal gravitational constant is $G = 6,67 \cdot 10^{-11}$. Express the force $F$ as a monomial in terms of $x$, knowing that $m_1 = 5 \cdot 10^5 x^3$, $m_2 = 2 \cdot 10^4 x^2$, and the distance separating them is $d = 10^2 x$. Simplify the coefficient using scientific notation.

Prove algebraically that if the product of the monomials $5x^{n+1} y^{m-1}$ and $x^{n-1} y^{m+1}$ is subtracted from the square of the monomial $3x^n y^m$, the result is always a monomial like $x^{2n} y^{2m}$. Determine exactly what this resulting monomial is.

Find a monomial $M(a,b)$ that satisfies the following algebraic equation, showing the detailed solving process:

$\frac{(M(a,b))^3}{2a^4b} = -32a^5b^8$

An open-top tank with the shape of a right prism with a square base is to be built. The side length of the base is $3x^2$ metres and the height is $\frac{1}{2}x$ metres. Express the total surface area of sheet metal required to build it (without the top lid) as a polynomial. Then, calculate the numerical value of this surface area if the volume of the tank is known to be $144$ cubic metres.

The monomial $A = \sqrt{18} x^3 y^4$ is multiplied by the monomial $B = \sqrt{2} x^5 y^2$. Express the product $A \cdot B$ as a simplified monomial, taking roots where possible. Then, find a monomial $C$ such that when performing the division $\frac{A \cdot B}{C}$ the result is exactly $3x^2 y$.

In a fluid dynamics study, the flow rate $Q$ through a pipe is mathematically modelled as the product of the cross-sectional area $S$ and the fluid velocity $v$. If the cross-section is circular with radius $r = \frac{1}{2}t^2$ and the velocity increases non-uniformly according to $v = \frac{8}{\pi}t^3$, obtain the monomial that defines the flow rate $Q$ in terms of time $t$. What is the degree of this resulting monomial?

[CHOOSE THE METHOD] To evaluate an algebraic expression in the laboratory, the numerical value of the following operation with monomials must be calculated for $x = -2$, $y = 3$, and $z = -1$:

Two students propose different paths to reach the solution:

• Alba's method: Substitute the values $x = -2$, $y = 3$, $z = -1$ directly into each of the three monomials in parentheses, calculate the numerical result of each factor, and finally multiply the three obtained numbers.

• Bruno's method: First multiply and simplify the three monomials by applying power properties to obtain a single reduced monomial (with its sign, fractional coefficient, and final exponents) and, only at the end, substitute the values of $x$, $y$, $z$.

1. a) Solve the exercise following Alba's method and then following Bruno's method, showing the intermediate steps in both.
2. b) Compare the two methods: which one minimizes the risk of making mistakes with negative signs and powers? Justify your answer by stating how many powers and sign changes you had to manage in each path.
3. c) If you had to program a spreadsheet to calculate this value for $100$ different experimental measurements of $(x, y, z)$, which of the two methods would you choose as the basis for the algorithm and why?

[MODEL] An express parcel and logistics company based in Zaragoza classifies its rates using algebraic models based on volume and weight. For a bulk shipment of cubic boxes of edge length $a$ (in meters), the internal handling cost $C$ (in euros) on the sorting conveyor belt fits the monomial:

where $N$ represents the total number of processed parcels, $a$ is the edge length of each cubic box (in meters), and $t$ is the average processing time per scanner (in seconds).

1. a) Interpret the physical meaning of the factor $a^3$ within the context of the problem and state the total degree of the monomial with respect to the set of its variables ($N$, $a$, $t$).
2. b) To reduce the costs of a campaign, the operations team considers two alternative options while keeping the scanner time $t$ constant:

• Option 1: Halve the size of the boxes, using edges of length $\frac{a}{2}$, transporting the same number $N$ of parcels.

• Option 2: Keep the original size of the boxes ($a$), but reduce the number of parcels to a quarter ($\frac{N}{4}$).

Prove algebraically, by substituting into the monomial formula, which of the two options achieves a greater reduction in cost $C$ and determine by what factor the cost is divided in each case.

1. c) A novice employee proposes adding the previous cost $C$ directly to the labeling cost $E(N, a, t) = 1,2 \cdot N \cdot a^2 \cdot t^3$ to obtain a single total cost monomial. Argue mathematically why this sum cannot be reduced to a single monomial and explain what practical consequence this has when quoting the service.