Exercises: Operations with monomials

In an adventure video game, your character collects energy gems. Red gems are worth $x$ points each. If during the first level you get $5x$ gems, in the second you find another $8x$, and in the third you lose $3x$ due to a spell, simplify the expression to find out how many gems you have at the end.

Calculate the perimeter of the following digital tablet knowing that its sides are expressed in terms of a variable $x$. Remember that the perimeter is the sum of all the sides of the rectangle.

To organize a party, you buy packs of soda. Each pack contains $6$ cans and the price of one can is $\$y$. If you buy $4$ identical packs, write the monomial that represents the total cost of the purchase.

Hint: First find out how many cans there are in total and then multiply by the price of one.

A farmer has a rectangular garden plot to plant tomatoes. Help him calculate the expression for the area of the plot by multiplying the following monomials:

Simplify the following expressions by combining like terms as much as possible:

• a) $2x + 7x + x$

• b) $9a - 5a + 2a$

• c) $10y^2 - 3y^2 - 4y^2$

• d) $3ab + 5ab - 8ab$

Perform the following multiplications of monomials, remembering that coefficients are multiplied together and the exponents of identical variables are added:

• a) $3 \cdot (4x)$

• b) $2x \cdot 5x$

• c) $(-3a^2) \cdot 2a$

• d) $4x \cdot 3y$

A YouTuber has received views on his latest video during the first three days. On the first day he had $12x^3$ views, on the second day $8x^3$, and on the third day $5x^3$. If he wants to share the total views from those three days among his $5$ channel moderators, what expression represents the views that correspond to each moderator?

Fill in the empty boxes with the necessary monomial so that the operation is correct:

In a popular adventure video game, your character has collected different items. If we call $x$ the health potions and $y$ the magic elixirs, simplify the following expression to find out the final inventory in your backpack:

$5x + 3y - 2x + 7y + x - 4y$

Calculate the perimeter of the following garden plot. Express the result as a simplified monomial in terms of the variable $x$.

Perform the following multiplications of monomials, remembering to add the exponents of identical variables:

1. a) $(4a^2) \cdot (3a^4) =$

1. b) $(-2x^3) \cdot (5x) =$

1. c) $(\frac{1}{2}b^2) \cdot (6b^5) =$

In a sports shop, a training pack contains $3$ weights and $2$ resistance bands. If the price of a weight is $\$x$ and the price of a band is $\$y$, express as a simplified monomial the total cost of buying four identical packs.

Simplify the following divisions of monomials:

1. a) $\frac{15x^6}{3x^2} =$

1. b) $\frac{-20a^4}{5a^3} =$

1. c) $\frac{8y^3}{8y^3} =$

Apply the distributive property to remove parentheses and combine like terms in the following expression:

$3 \cdot (2x + 4) + 5 \cdot (x - 2)$

Find the expression for the area of the following rectangle as a monomial. Remember that the area is calculated by multiplying the base by the height ($A = b \cdot h$).

Find the mystery monomial missing in each of the following equations so that the statement is correct:

1. a) $7x^2 +$ $\text{\_\_\_\_\_\_}$ $= 12x^2$

1. b) $10a^5 :$ $\text{\_\_\_\_\_\_}$ $= 2a^2$

1. c) $\text{\_\_\_\_\_\_}$ $\cdot (3x) = 18x^4$

A popular streamer is analyzing the statistics of his channel over a week. On Monday he received $5x$ new subscribers and $3y$ comments. On Tuesday, his subscribers tripled compared to Monday, but the comments fell to a third of those on Monday. On Wednesday, he lost $2x$ subscribers and received twice as many comments as on Monday.

Write and simplify the algebraic expression that represents the total number of subscribers and comments accumulated over those three days. Is it possible to add the terms with $x$ and with $y$? Justify your answer using the concept of like terms.

Look at the design of this play area for a technology park. The local council needs to calculate the total area to install artificial turf of different colors in each section.

Calculate, using operations with monomials, the following expressions:

1. The area of Zone A.

2. The area of Zone B (formed by two identical triangles).

3. The total area of the enclosure by simplifying the sum of the resulting monomials.

In the math lab, we have created a "Monomial Multiplication Pyramid." In this pyramid, the value of each block is obtained by multiplying the two monomials directly below it. Fill in the empty boxes by performing the necessary operations:

A delivery drone makes a round trip to deliver a package. The outward distance is given by the monomial $D(x) = 12x^2$. Due to a headwind, the average outward speed is $v_1 = 4x$. On the return trip, without a load and with a tailwind, the average speed is twice the outward speed.

Knowing that time is calculated as $t = \frac{\text{Distance}}{\text{Speed}}$, perform the following operations:

1. Divide the corresponding monomials to find the outward time.

2. Find the return time by performing the relevant division.

3. Express the total trip time as a single simplified monomial.

Look at the following series of figures built with white and gray square tiles. Mathematicians call this "growth patterns."

1. If $n$ represents the figure number, write a monomial that represents the number of gray tiles in figure $n$.

2. Write a monomial for the number of white tiles in figure $n$.

3. Add both monomials to obtain the expression for the total number of tiles. Which figure would have a total of 101 tiles?

A "Mystery Monomial" has lost some of its parts after passing through an operations machine. Your mission is to reconstruct the original equation by finding the missing monomial in each case (represented by the $?$ sign):

1. a) $? + 7a^2b - 3a^2b = 10a^2b$

1. b) $(? \cdot 5x^4) : 2x^2 = 15x^5$

1. c) $\frac{24y^{10}}{?} = 4y^5$

A video game designer uses monomials to program the strength of an attack. The base power is $P = 4ab^2$. If the character uses a "fury potion," their power is multiplied by $3a^2$. If they then receive a "curse of weakness," the previous result must be divided by $6ab$.

Perform the multiplication and division operations of monomials to find the simplified expression of the final attack power. State the coefficient and the degree of the resulting monomial.

Simplify the following expression, paying close attention to the order of operations and the properties of exponents:

Explain step by step how you reduced each term before performing the final addition and subtraction. What is the degree of the resulting polynomial before total simplification?