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.

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.

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 operating with monomials and reduce the result to its simplest form:

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

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

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) $(4x^2) \times (3x^5)$

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

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

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

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.

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.

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

$$C(x, y) = 3x^2 y^3 + 2(x y)^2 \times y - (2x y^2)^2$$

On her exam, Aoife writes:

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

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 Aoife 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.

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.

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:

$$L(x, y) = \frac{(2x^3y^2)^3 \times (\frac{1}{2}x^2y^2)^2}{\sqrt{16x^8y^{10}}} : (x^2y^2)^2$$

State the degree of the resulting monomial.

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.

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?

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?