Monomials

In the world of algebra, not all expressions are "monomials". Remember that a monomial is a product of a number and a letter (or several) with non-negative integer exponents. Indicate which of the following expressions are monomials and which are not. If not, explain why:

1. a) $5x^3$
2. b) $2 + a$
3. c) $-3ab^2$
4. d) $\frac{4}{x}$
5. e) $x$
6. f) $\sqrt{x}$

Copy the following table into your notebook and complete the missing elements for each monomial. Be careful with signs and invisible exponents!

In a strategy video game, resources are stored using algebraic expressions. Complete the following table so that the programmer can organize the inventory correctly:

Two monomials are "like terms" if they have exactly the same variable part. In the following list, there are three hidden pairs of like monomials. Find them and group them!

• $4x^2$

• $-5ab$

• $\frac{1}{3}x$

• $2x^2$

• $8ab$

• $-7x$

Simplify these expressions by adding or subtracting the monomials that are like terms. If an expression cannot be simplified, explain why:

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

You are organizing the files on your computer. To save space, you can only add or subtract those "files" (monomials) that are like terms. Reduce the following expression as much as possible:

$7x^2 - 3x + 2x^2 - 10x + 5$

In the comic book shop, the price of a manga is $\$x$. If Ethan buys three times the square of the number of mangas and then gets a discount of twice the square of mangas, what monomial represents the final number of mangas he paid for? Write the operation and simplify.

We want to fence a dog run at an animal shelter. The enclosure has the shape of a rectangle. The long side is three times the short side. If we call $x$ the short side:

1. a) Write a monomial for the long side.
2. b) Write an expression for the total perimeter of the enclosure and simplify it to obtain a single monomial.

A shelf with a special polygonal shape is to be surrounded with an LED strip. Calculate the monomial that represents the total perimeter of the figure by adding its sides:

Calculate the result of the following multiplications of monomials. Remember to multiply the numbers and the variables using the rules of exponents:

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

Calculate the product of these two monomials that represent the power of two data servers:

$P_1 = -4a^2b^3$

$P_2 = \frac{1}{2}ab^2$

Remember to correctly apply the rules of signs and the exponent rules for multiplying terms with the same base.

An influencer wants to calculate the area of their new smartphone screen to design a custom wallpaper. If the base of the phone measures $3x$ in and the height measures $5x^2$ in, write the monomial that represents the total area.

A streaming server divides a data packet of size $24x^6$ among $4x^2$ users. Write the monomial that represents the amount of data each user receives.

Complete the following "Monomial Pyramid". In this pyramid, the value of each box is the product of the two monomials below it. You must find the missing monomials at the base so that the monomial at the top is correct.

We are organizing a video game tournament. The in-game currency is "Gems" ($g$) and "Gold Coins" ($m$). Express the following situations using a monomial:

1. a) Three times the gems you have.
2. b) Half of the gold coins.
3. c) The price of buying $5$ packs if each pack costs $x$ gems.
4. d) The area of a square rug for your virtual room whose side measures $s$.

Imagine that an influencer earns $\$5x$ for each sponsored post they make on their social media, where $x$ is the number of days the ad is visible.

1. a) What monomial represents what they earn if they make $3$ identical posts?
2. b) Calculate the numerical value (the actual earnings) if the ad is visible for $x = 10$ days.

Find the numerical value of the monomial $V(x, y) = -3x^2y$ for the following values of the variables (scores in a game):

1. a) For $x = 2$ and $y = 5$.
2. b) For $x = -1$ and $y = -4$.

Calculate the numerical value of the resulting expression from the following operation with monomials for $x = -2$ and $y = 3$:

Hint: Simplify the expression first using the properties of powers before substituting the values.

An urban artist is designing a mural with interlaced geometric shapes. The central design consists of a large rectangle with base $5x^2$ and height $3y$. Inside this rectangle, the artist leaves an unpainted gap in the shape of a triangle whose base measures $2x^2$ and its height is $y$. Write the simplified monomial that represents the area that will finally be painted in color.

Observe the following tower built with decorative cubes. Each cube has an edge that measures $a$ in. It is known that the top face of each cube is painted gold, and the outer side faces (which are not touching another cube) are painted silver.

1. Write the monomial that represents the total area painted gold in a tower of $n$ floors.

2. Write the monomial that represents the total area painted silver if the tower has exactly 4 floors like the one in the image.

In a strategy video game, the damage a spell inflicts depends on the wizard's level ($x$) and their magical energy ($y$). The basic fire spell has a value of $6x^2y^3$. If a wizard gets a power-up that multiplies their damage by $\frac{2}{3}xy^2$ and then the enemy reduces that total damage to a quarter due to their armor, what is the monomial that represents the final damage received by the enemy? Also indicate the degree of the resulting monomial.

A logistics company uses drones to deliver parcels. The battery consumption of a drone on a round trip is given by the expression $8ab^2$. If on a very windy day the consumption increases by $3ab^2$ and, in addition, due to carrying a heavy load the total consumption doubles, what algebraic expression would represent the consumption of 5 drones making the same journey under those conditions?

An influencer analyses their weekly statistics. They have observed that the number of likes they receive is twice their followers ($s$) multiplied by the square root of their posts ($p$), but since the square root is difficult to use, their team uses the monomial $5s^2p$ to estimate the reach. If this week they have had $3s^2p$ photo likes and $7s^2p$ video likes, and the system mistakenly deletes three times the difference between videos and photos, write the simplified monomial that represents their final likes.

Invent two monomials, $M_1$ and $M_2$, that meet all these conditions at the same time:

1. Both must have $a^2b$ as their literal part.

2. Their sum must be a monomial whose coefficient is $-10$.

3. When dividing $M_1$ by $M_2$, the result must be the number $4$.

Write the two monomials and check that they meet the conditions.