Exercises: Operations with monomials

Calculate the result of the following addition and subtraction of like monomials:

$3x^2y - 5x^2y + \frac{1}{2}x^2y$

Multiply the following monomials and simplify the result:

$(2ab^2) \cdot (-3a^2b) \cdot (4a)$

Perform the division of the following monomials:

$\frac{-15x^4y^3z}{5x^2y}$

Calculate the power of the following monomial:

$(-2x^3y^2)^4$

Identify which of the following monomials are like terms and simplify the expression:

$4m^3n - 2mn^3 + 5m^3n + mn^3$

Calculate the numerical value of the monomial $-3x^2y^3$ for the values $x = 2$ and $y = -1$.

Find the missing monomial in the following operation so that the equality is true:

$7x^3 + (\dots) = -2x^3$

Determine which monomial $4x^2$ must be multiplied by to obtain $-20x^5$ as a result.

In a 4th-year ESO class, Marcos has to simplify the following combined operation with monomials: $5x^3 + 3x^3 - (2x^2)^3$. On his exam, he writes the following step-by-step working:

• Step 1: $5x^3 + 3x^3 = 8x^6$

• Step 2: $(2x^2)^3 = 6x^5$

• Step 3: $8x^6 - 6x^5 = 2x$

Complete the following tasks:

1. a) Identify the three mistakes made by Marcos and explain for each one which property of operations with monomials (addition, power, or subtraction) he applied incorrectly.
2. b) Correct the exercise in your notebook by solving it step by step properly, indicating the order of operations you followed.

A renewable energy company designs a new ground-mounted photovoltaic installation in Extremadura. Each standard solar panel has a width of $x$ meters and a length of $1,8x$ meters.

1. a) Calculate the monomial that expresses the area of a single solar panel in terms of the variable $x$.
2. b) The project includes placing a continuous row of $250$ identical panels. Obtain the simplified monomial representing the total collection area of that row by multiplying the number of panels by the monomial of the unit area.
3. c) The technical specifications require leaving a perimeter maintenance and cooling strip whose area is given by the monomial $110x^2$. Explain why it is possible to add the collection area and the maintenance area into a single monomial using the concept of like terms, and find the monomial representing the total area occupied by the installation.

Perform the following combined operations with monomials, respecting the order of operations:

$2x \cdot (3x^2)^2 - 5x^2 \cdot (x^3)$

A rectangular plot of land has dimensions given by monomials. Its base measures $4xy^2$ and its height $3x^2y$. Express the area and perimeter of the plot using simplified algebraic expressions.

In physics, the kinetic energy of an object is calculated using the formula $E_c = \frac{1}{2} m v^2$. If the mass of a vehicle can be expressed as $m = 4x$ and its speed as $v = 3xy$, express its kinetic energy as a simplified monomial.

Perform the following combined operation with monomials involving fractional coefficients and simplify the result:

$\left( \frac{2}{3} a^2b \right)^3 : \left( \frac{4}{9} a^4b^2 \right)$

A water tank is cylindrical in shape. The radius of its base is $r = 2x^2$ and its height is $h = 5xy$. Knowing that the volume of a cylinder is $V = \pi r^2 h$, express the volume of the tank as a monomial.

A technology company sells electronic components. The selling price of each component is given by the monomial $P = 15x^2y$ euros. If in one month they sell a quantity of components equivalent to $Q = 4xy^3$, express the company's total revenue as a monomial.

The base of a triangle measures exactly three times its height $h$. Express the area of this triangle as a monomial in terms of the variable $h$.

Subtract the monomial $5a^3b^4$ from the product of the monomials $-2ab^2$ and $3a^2b^2$. Express the final simplified result.

To simplify the expression with monomials $\left(\frac{6x^4y^3}{2x^2y}\right)^3$, two students propose different approaches:

• Alba's method: First computes the quotient of monomials inside the parentheses and, then, raises the resulting monomial to the third power.

• Bruno's method: First applies the power of a quotient property by raising the numerator and denominator to the third power (expanding the powers of monomials) and, finally, carries out the division of the resulting monomials.

1. a) Solve the expression following Alba's proposed method step by step.
2. b) Solve the expression following Bruno's proposed method step by step and verify that both results match.
3. c) Compare both approaches by assessing the size of the numerical coefficients and the difficulty of the intermediate exponents. Justify which of the two methods you consider more convenient to minimize the risk of calculation errors.

An olive oil cooperative in Jaén stores extra virgin olive oil in cylindrical steel tanks. The volume of a cylindrical tank is calculated using the formula $V = \pi \cdot r^2 \cdot h$, where $r$ is the base radius and $h$ is the height. In their standard models, the engineers set a radius of $r = x$ meters and a height of $h = 2,5x$ meters (using the approximation $\pi \approx 3,14$).

1. a) Multiply the coefficients and the powers of the variable to obtain the simplified monomial that represents the volume $V_1(x)$ of a standard tank.
2. b) If the oil mill has a row of $8$ standard tanks, write the simplified monomial that expresses the total storage capacity of that row.
3. c) To expand capacity, the cooperative orders a giant prototype in which the base radius is doubled ($r = 2x$) and the height is tripled ($h = 7,5x$). Expand the monomial describing the volume $V_2(x)$ of the new prototype and calculate the quotient of monomials $\frac{V_2(x)}{V_1(x)}$ to determine, with reasoning, exactly how many standard tanks a single giant tank is equivalent to.

Find the monomial $M(x,y)$ that satisfies the following algebraic equation:

$\frac{(3x^2y)^2 \cdot M(x,y)}{-2xy^3} = 18x^5y^2$

A logo design consists of an outer square of side length $4x^2y$ containing an inner square of side length $3x^2y$. Express the area of the region between the two squares (the shaded area) as a simplified monomial.

An open-topped box is to be constructed from a rectangular cardboard sheet measuring $10x$ by $8x$. To do this, a square of side length $x$ is cut out from each of the four corners and the flaps are folded up. Prove that the volume of the resulting box is a monomial and calculate it.

Newton's law of universal gravitation states that the gravitational force between two masses is $F = G \frac{m_1 m_2}{d^2}$. If the mass of the first body is $m_1 = 2x^2$, that of the second is $m_2 = 8x^3$, and the distance separating them is $d = 4x$, express the force $F$ as a monomial in terms of the constant $G$ and the variable $x$.

Prove algebraically that if you take a cube with edge length $2a^2$, square its volume and divide the result by the area of one of its faces, you always get a monomial. Work out what that monomial is.

Fully simplify the following nested expression with monomials, applying index laws and operations with monomials:

$\left[ \frac{(-2x^3y^2)^3 \cdot (x^2y)^2}{(4x^5y^3)^2} \right]^2$

A textile factory produces batches of T-shirts. The cost of manufacturing a batch of $N = 5x^2$ T-shirts is $C = 20x^3$ euros per T-shirt. Subsequently, all manufactured T-shirts are sold at a price of $V = 35x^3$ euros each. Express the total profit obtained by the factory (Revenue minus Costs) as a simplified monomial.

The volume of a cone is given by $V_c = \frac{1}{3}\pi r^2 h$ and that of a sphere by $V_e = \frac{4}{3}\pi R^3$. A cone has a base radius of $r = 2x$ and a height of $h = 4x$. A sphere has a radius of $R = x$. Calculate the ratio (the quotient) of the volume of the cone to the volume of the sphere, showing that the result is an integer independent of $x$.

A regional wastewater treatment plant manages sewage treatment using a drive and filtration system. Technicians model the treated flow rate (in cubic meters per hour) through the following combined operation with monomials:

where $x$ is the inlet pressure in bars, $y$ is the diameter of the pipes in decimeters, and $M(x)$ is an unknown monomial representing the performance of the main motor (which depends solely on the pressure $x$).

1. a) For water to flow at a safe speed without wearing down the membranes, engineers establish that the resulting flow rate must be exactly equal to the monomial $\frac{12x^5}{y}$ (or expressed with an exponent: $12x^5y^{-1}$). Apply the rules of product, power, and quotient of monomials to isolate and deduce the algebraic expression of the monomial $M(x)$.
2. b) State the coefficient, the literal part, and the degree of the monomial $M(x)$ obtained, and verify that substituting it into the original expression satisfies the equality.
3. c) If due to a breakdown the motor is temporarily replaced by another whose performance is given by the monomial $M'(x) = 4x^2$, calculate the ratio between the original calibrated flow rate and the new flow rate to justify by what factor the water treatment rate is reduced.

According to technical data from Red Eléctrica de España on wind energy, the theoretical power in watts ($W$) generated by a wind turbine is approximated by the monomial $P = 0,6 \cdot S \cdot v^3$, where $S$ is the circular swept area of the blades ($S \approx 3,14 \cdot R^2$, with $R$ being the length of each blade) and $v$ is the incident wind speed. In a wind farm, the design of the turbines sets blade length to $R = 2x$ meters and an average wind speed modeled as $v = 1,5x$ meters per second.

1. a) Estimate before calculating: If due to atmospheric or design conditions the scale variable $x$ is multiplied by $2$ (doubled), by what approximate factor do you intuitively think the power $P$ will be multiplied: by 2, by 4, by 8, or by more than 20? Justify your prior estimate by looking at the exponents that will affect the variable $x$ in each factor of the formula.
2. b) Algebraically perform all the power and product operations with monomials that make up the power:

Write the resulting monomial by rounding its coefficient to two decimal places and specify its total degree.

1. c) Verify and reflect: Using the obtained monomial, calculate the quotient $\frac{P(2x)}{P(x)}$ by algebraically substituting $x$ with $2x$. Compare the resulting exact factor with the estimate you made in section a) and explain the practical implications this result has regarding the importance of installing wind turbines in areas with slightly stronger winds.