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 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) $4x^3 - 5x + 2x^3 + 9x$

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.

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.

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).

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.

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.

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.

Consider the monomial $M(a, b) = k \times a^{n+1} \times b^{2m-3}$. 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 $m$.

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 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 pesos).

1. Calculate the numerical value of the profit if the company has $20,000$ users ($x=20$) and the marketing spend is $200$ pesos per user ($y=200$).

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 center, the volume of air needed to cool a server rack is modeled 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?

It is desired to distribute a government grant of $S$ pesos among a number of municipalities $N$ that depends on the population. The total grant is defined by the monomial $S(a, b) = 2\,880a^6b^4$ and the number of municipalities by $N(a, b) = (2a^2b)^2$.

Calculate the monomial that represents the amount in pesos 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?