Monomials

In an app store, the profit for each download of a game is expressed by the monomial $7x^2y$. Identify the following elements of this expression:

• Coefficient:

• Literal part:

• Degree of the monomial:

In a math app, a bot has misclassified some monomials. Your task is to correct the following table by identifying the coefficient, the literal part and the degree of each one. If you find any errors in the bot's classification, write it correctly:

Monomial	Coefficient	Literal part	Degree
$-5x^2y$	$5$	$x^2y$	$2$
$\frac{3}{4}ab^3c$	$3$	$ab^3c$	$4$
$-x^4$	$-1$	$x$	$4$
$7$	$7$	None	$1$

We are organizing an inventory of robotics parts. Group the following monomials into pairs that are like terms:

$5x^2$, $-3ab$, $x$, $\frac{1}{2}ab$, $-2x^2$, $8x$

Calculate the numerical value of the monomial $M(x) = -5x^2$ for the indicated values of the variable:

1. a) $x = 2$
2. b) $x = -1$
3. c) $x = 0$

Calculate the numerical value of the following monomials for the values of the variables indicated in each case:

1. a) $P(x, y) = -3x^2y$ for $x = -2$, $y = 5$

1. b) $Q(a, b) = \frac{2}{3}a^3b^2$ for $a = 3$, $b = -1$

Reduce the following algebraic expressions by adding and subtracting the like terms:

1. a) $4x + 2x - x$
2. b) $5a^2 - 3a^2 + 7a^2$
3. c) $3ab + ab - 2ab$

A content creator analyzes the growth of their social networks. The number of "Likes" on their photos is given by the monomial $8x^2y$, and the number of "Shares" by $3x^2y$. During the last month, they have received extra views equivalent to $\frac{1}{2}x^2y$.

Calculate the simplified algebraic expression that represents the total interaction by adding the three concepts.

Perform the following multiplications of monomials and simplify the result:

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

In a strategy video game, your base's energy increases according to the following operation of monomials. Reduce the expression to a single monomial:

$$(12x^5 \cdot 3x^2) : (9x^4)$$

What is the degree of the resulting monomial?

Simplify the following expression that combines powers and divisions of monomials:

$$\frac{(-2a^3b)^2 \cdot 5ab^4}{10a^4b^3}$$

Simplify the following combined expression with monomials as much as possible, applying the properties of powers and the order of operations:

$$\frac{(-3x^2y^3)^2 \cdot (2x^4y)^3}{(-6x^3y^2)^2}$$

Express the result as a single monomial.

Calculate the degree of this "chained" monomial and simplify it to its simplest form:

$$P(x) = x \cdot (2x^2)^3 : \left( \frac{1}{4}x^4 \right) + (-5x^3) \cdot (-2x^0)$$

Are the terms resulting from the sum like terms? Justify your answer.

Complete the gaps with the necessary monomial so that the equalities are correct:

1. a) $7x^3y + \dots \dots = 10x^3y$

1. b) $(-2ab^2) \cdot (\dots \dots) = 12a^3b^2$

1. c) $\frac{15x^4y^2}{\dots \dots} = 3x^2y$

In the following "product pyramid", each brick is the result of multiplying the two monomials directly below it. Complete the empty spaces by finding the monomials $A$, $B$ and $C$.

It is known that the numerical value of the monomial $M(x, y) = -5x^a y^b$ is $-160$ when $x=2$ and $y=2$. Furthermore, it is known that the relative degree with respect to the variable $x$ is twice the relative degree with respect to $y$.

Find the exponents $a$ and $b$, justifying the steps taken.

Translate the following phrases into algebraic language using a single monomial:

1. a) Twice the cube of a number $x$.
2. b) Half the product of two numbers $a$ and $b$.
3. c) Three times the square of the distance $d$.

Observe the following regular hexagon whose side measures $x$ feet. Write the monomial that represents its perimeter and calculate its value if each side measured $4,5$ inches.

Find the monomial that expresses the area of the following rectangle intended for a "gaming" zone in a shopping mall:

Find the monomial that represents the area of the following shaded zone, which is part of the design of a new skate park. Express the result in the most simplified form possible.

Calculate the monomial that represents the perimeter and the monomial that represents the shaded area of the following composite figure (a large square of side $3x$ from which four identical small squares of side $y$ have been cut out at its corners):

Reduce the expressions as much as possible.

Observe the following figure composed of two adjoining rectangles.

1. a) Write the monomial that represents the area of each rectangle separately.
2. b) Write the monomial that represents the total area.
3. c) Extract the common factor in the expression for the total area.

Extract the greatest possible common factor (greatest common factor of the terms) in the following algebraic expression. Pay special attention to the fractional coefficients:

$$\frac{4}{9}a^5b^2 - \frac{8}{27}a^3b^4 + \frac{2}{3}a^3b^2$$

Three friends rent a pickleball court. The total cost is given by the algebraic expression $C(t, l) = 6t + 2l$, where $t$ is the time in hours and $l$ is the number of paddles rented.

1. a) If they decide to divide the cost equally, write the algebraic expression that represents what each one must pay.

1. b) If a fourth friend joins the group, the total cost of the paddles increases (there are now $l+1$) and the cost is divided by four. Write the new expression for the individual payment.

1. c) If the playing time is $1,5$ hours and each one rents their own paddle, how many dollars does each of the three original friends save when the fourth player joins?

Observe the following sequence of figures formed by small squares. The first term ($n=1$) has 4 squares, the second ($n=2$) has 7, and the third ($n=3$) has 10.

1. a) Determine the monomial that represents the number of squares in terms of the position $n$, knowing that the relationship is linear (a monomial in $n$ plus a constant term).

1. b) How many squares will the figure in position $n=150$ have?

1. c) If a figure has 91 squares, what position $n$ does it occupy in the sequence?