Operations with Monomials
In a video game, each red gem is worth $x$ points. In level 1 you earn $5x$ points, in level 2 you earn $8x$, and in level 3 a spell takes away $3x$. How many points do you have in total? To find out, you need to work with monomials.
Monomials and like terms
An expression like $5x$ or $-3a^2$ is a single "block" made of a number and some letters. These blocks are called monomials.
Remember that an exponent counts how many times a letter is multiplied: $x^3 = x \cdot x \cdot x$.
Not everything with letters is a monomial. $\frac{4}{x}$ is not one, because the letter is in a denominator. $2 + a$ is not one either, because it has two terms joined by a $+$.
Two monomials are like terms if they have exactly the same variable part. Think of fruit: 3 apples + 2 apples = 5 apples, but 3 apples + 2 oranges cannot be merged into one kind of fruit. In the same way, $4x^2$ and $2x^2$ are like terms, but $4x^2$ and $4x$ are not.
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…
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.
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) $3ab + 5ab - 8ab$
Simplify the following expressions by combining like terms as much as possible:
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a) $2x + 7x + x$
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b) $9a - 5a + 2a$
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c) $10y^2 - 3y^2 - 4y^2$
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d) $3ab + 5ab - 8ab$
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…
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:
$5x + 3y - 2x + 7y + x - 4y$
Perimeter with monomials
The perimeter of a shape is the sum of the lengths of all its sides. When the sides are monomials, perimeter is a perfect place to combine like terms.
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…
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.
Calculate the perimeter of the following garden plot. Express the result as a simplified monomial in terms of the variable $x$.
Calculate the perimeter of the following garden plot. Express the result as a simplified monomial in terms of the variable $x$.
Multiplying monomials
Adding is about counting blocks. Multiplying is different: letters get multiplied too, and that changes their exponents.
When the letters are different, they just sit side by side: $4x \cdot 3y = 12xy$. A plain number counts as a coefficient: $4 \cdot 6y = 24y$.
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…
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.
Perform the following multiplications of monomials, remembering that coefficients are multiplied together and the exponents of identical variables are added: a)…
Perform the following multiplications of monomials, remembering that coefficients are multiplied together and the exponents of identical variables are added:
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a) $3 \cdot (4x)$
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b) $2x \cdot 5x$
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c) $(-3a^2) \cdot 2a$
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d) $4x \cdot 3y$
Perform the following multiplications of monomials, remembering to add the exponents of identical variables: a) $(4a^2) \cdot (3a^4) =$ b) $(-2x^3) \cdot (5x) =$ c)…
Perform the following multiplications of monomials, remembering to add the exponents of identical variables:
- a) $(4a^2) \cdot (3a^4) =$
- b) $(-2x^3) \cdot (5x) =$
- c) $(\frac{1}{2}b^2) \cdot (6b^5) =$
Area problems
The area of a rectangle is base times height, $A = b \cdot h$. When the base and height are monomials, you multiply monomials. If the sides are measured in feet, the area is in square feet.
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:
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:
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 expression for the area of the following rectangle as a monomial. Remember that the area is calculated by multiplying the base by the height ($A = b \cdot h$).
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…
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:
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The area of Zone A.
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The area of Zone B (formed by two identical triangles).
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The total area of the enclosure by simplifying the sum of the resulting monomials.
Dividing monomials
Division undoes multiplication, so it undoes what we did to the exponents too. Some problems write division with a colon: $a : b$ means $a \div b$.
Coefficients do not always divide evenly. Then write a fraction: $\dfrac{10x^4}{4x^2} = \dfrac{10}{4}x^2 = \tfrac{5}{2}x^2$.
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…
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?
Simplify the following divisions of monomials: a) $\frac{15x^6}{3x^2} =$ b) $\frac{-20a^4}{5a^3} =$ c) $\frac{8y^3}{8y^3} =$
Simplify the following divisions of monomials:
- a) $\frac{15x^6}{3x^2} =$
- b) $\frac{-20a^4}{5a^3} =$
- c) $\frac{8y^3}{8y^3} =$
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…
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:
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Divide the corresponding monomials to find the outward time.
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Find the return time by performing the relevant division.
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Express the total trip time as a single simplified monomial.
Combined expressions
Real problems mix parentheses, sums, products, and powers. The order of operations still rules: parentheses and exponents first, then multiplication and division from left to right, and finally addition and subtraction.
A power of a monomial means repeated multiplication: $(2x^3)^2 = 2x^3 \cdot 2x^3 = 4x^6$. Square the coefficient and double the exponent.
The distributive property lets you remove parentheses: $a(b + c) = ab + ac$. Picture it as a rectangle split into two pieces.
The degree of an expression with several terms is the highest degree among its terms. In the example above, every term has degree 4.
Apply the distributive property to remove parentheses and combine like terms in the following expression: $3 \cdot (2x + 4) + 5 \cdot (x - 2)$
Apply the distributive property to remove parentheses and combine like terms in the following expression:
$3 \cdot (2x + 4) + 5 \cdot (x - 2)$
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…
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: $$\frac{20x^6}{5x^2} + (3x^2 \cdot 4x^2) -…
Simplify the following expression, paying close attention to the order of operations and the properties of exponents:
$$\frac{20x^6}{5x^2} + (3x^2 \cdot 4x^2) - \frac{(2x^3)^2}{x^2}$$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?
Mystery monomials and puzzles
Some problems hide a monomial and ask you to find it. The trick is to work backward using the inverse operation.
| Operation with the mystery | Undo it with |
|---|---|
| $? + A = B$ | $? = B - A$ |
| $? \cdot A = B$ | $? = B \div A$ |
| $? \div A = B$ | $? = B \cdot A$ |
| $A \div\, ? = B$ | $? = A \div B$ |
Why? It is the same idea as solving $\square + 3 = 10$ with numbers. You undo what was done.
The same idea works in a multiplication pyramid: each block is the product of the two blocks right below it. Going up, you multiply. If a block in the middle is missing, you go down and divide.
Check the pyramid: $2x \cdot 3x = 6x^2$, $3x \cdot 4 = 12x$, and $6x^2 \cdot 12x = 72x^3$.
Fill in the empty boxes with the necessary monomial so that the operation is correct:
Fill in the empty boxes with the necessary monomial so that the operation is correct:
Find the mystery monomial missing in each of the following equations so that the statement is correct: a) $7x^2 +$ $\text{\\\\\\}$ $= 12x^2$ b) $10a^5 :$ $\text{\\\\\\}$…
Find the mystery monomial missing in each of the following equations so that the statement is correct:
- a) $7x^2 +$ $\text{\_\_\_\_\_\_}$ $= 12x^2$
- b) $10a^5 :$ $\text{\_\_\_\_\_\_}$ $= 2a^2$
- c) $\text{\_\_\_\_\_\_}$ $\cdot (3x) = 18x^4$
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…
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 "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…
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):
- a) $? + 7a^2b - 3a^2b = 10a^2b$
- b) $(? \cdot 5x^4) : 2x^2 = 15x^5$
- c) $\frac{24y^{10}}{?} = 4y^5$
Word problems and patterns
In word problems, first decide what the letters stand for. Then translate each sentence into a monomial: "doubled" means multiply by $2$, "a third of" means divide by $3$, "lost" means subtract.
Notice the final answer has two terms. Terms with $x$ and terms with $y$ are not like terms, so they cannot be merged into one.
Letters also describe patterns. Suppose figure $n$ is built with $n$ gray tiles on top and $2n$ white tiles on the bottom.
Gray tiles: $n$. White tiles: $2n$. Total tiles: $n + 2n = 3n$. For figure $n = 10$ you need $3 \cdot 10 = 30$ tiles. Combining like terms turns a long count into one short formula.
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…
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.
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…
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 following series of figures built with white and gray square tiles. Mathematicians call this "growth patterns." 1. If $n$ represents the figure number, write…
Look at the following series of figures built with white and gray square tiles. Mathematicians call this "growth patterns."
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If $n$ represents the figure number, write a monomial that represents the number of gray tiles in figure $n$.
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Write a monomial for the number of white tiles in figure $n$.
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Add both monomials to obtain the expression for the total number of tiles. Which figure would have a total of 101 tiles?
Back to the start: In the video game, you combine the like terms from the three levels: $5x + 8x - 3x = (5 + 8 - 3)x = 10x$. You finish with $10x$ points.