Exercises: Monomials

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

• Coefficient:

• Literal part:

• Degree of the monomial:

We are organising an inventory of robotics parts. Group the following monomials into pairs of 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) $x = 0$

Look at the following regular hexagon whose side measures $x$ metres. Write the monomial representing its perimeter and calculate its value if each side were $4.5$ cm.

Simplify the following algebraic expressions by adding and subtracting like monomials:

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

Perform the following monomial multiplications and simplify the result:

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

Find the monomial expressing the area of the following rectangle intended for a gaming area in a shopping centre:

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

1. a) Double 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 $d$.

Aoife has to simplify the expression $5x^3 + 3x^3$ in a class exercise and writes on the board: $5x^3 + 3x^3 = 8x^6$. When asked by the teacher, she justifies her answer saying: "I added the coefficients ($5 + 3 = 8$) and I also added the exponents of the letters ($3 + 3 = 6$)".

1. a) Identify the mistake Aoife made and explain, using the distributive property (factoring out the literal part), why the exponents should not be added.
2. b) Write the correct result of the addition.
3. c) What operation between the monomials $5x^3$ and $3x^3$ would have had to be performed for the result to be a monomial of degree $6$? Justify your answer.

We want to construct a monomial that simultaneously satisfies the following three conditions:

1) Its variables are $a$ and $b$, and its total degree is $4$.

2) The exponent of variable $b$ is exactly three times that of variable $a$.

3) Its coefficient is a negative integer and its numerical value for $a = 2$ and $b = -1$ is equal to $10$.

1. a) Reason step by step what the exponents of $a$ and $b$ must be.
2. b) Set up the equation that allows you to deduce the coefficient of the monomial from the given numerical value.
3. c) Write the final algebraic expression of the monomial and check that it verifies all the conditions.

In a maths app, a bot has incorrectly classified 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 error 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	$1$

A content creator is analysing the growth of their social media. 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 received an extra number of views equivalent to $\frac{1}{2}x^2y$.

Calculate the simplified algebraic expression representing the total interaction by adding the three items.

Find the monomial representing the area of the following shaded region, which is part of the design of a new skatepark. Express the result in its simplest form.

Work out the numerical value of the following monomials for the values of the variables given 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 $a = 3$, $b = -1$

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

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

What is the degree of the resulting monomial?

Simplify the following expression combining powers and divisions of monomials:

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

Look at the following figure formed by two adjacent rectangles.

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

Fill in the gaps with the required monomial to make the equalities correct:

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

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

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

Given the monomial $P(a, b) = -4a^2b^3$, we want to calculate its numerical value for $a = -2$ and $b = -1$.

1. a) Estimate the sign: Before performing any full numerical calculation, determine with reasoning whether the final result will be positive or negative. To do this, analyze separately the sign of the coefficient $-4$, of the power $(-2)^2$, and of the power $(-1)^3$.
2. b) Estimate the order of magnitude: Without calculating the exact value, deduce whether the absolute value of the result will be greater or less than $100$. Justify your mental estimation.
3. c) Check: Substitute the values into the monomial, calculate the final result step by step respecting the order of operations, and confirm whether your predictions from the previous sections were correct.

In a glass recycling plant in County Cork, a crusher stores the material in cubic hoppers whose edge measures $x$ metres. The mass of crushed glass (in kilograms) that fits into a full hopper is given by the monomial $M(x) = 1500x^3$.

1. a) Interpret and explain the physical meaning in this real-world context of each part of the monomial: the coefficient $1500$, the variable $x$, the power $x^3$, and the degree of the monomial.
2. b) If the plant currently uses hoppers with an edge of $x = 0.8\text{ m}$, calculate how many kilograms of crushed glass a full hopper contains.
3. c) In a working day, $5$ of these hoppers with edge $x$ are completely filled. Express the total mass collected in the day using a single simplified monomial and justify why it was possible to combine these quantities.

Observe the following sequence of patterns made of 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 expression that represents the number of squares in terms of position $n$, knowing that the relationship is linear (a monomial in $n$ plus a constant term).

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

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

Simplify the following expression involving monomials as far as possible, applying the laws of indices and the order of operations:

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

Express the result as a single monomial.

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

Simplify the expressions as much as possible.

Factorise the following algebraic expression by taking out the highest common factor of the terms. Take special care with fractional coefficients:

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

In the following "product pyramid", each brick is the result of multiplying the two monomials directly below it. Complete the missing 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 $-320$ 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 followed.

Three friends hire a padel 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 hired rackets.

1. a) If they decide to split the cost equally, write the algebraic expression representing what each person has to pay.

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

1. c) If the playing time is $1.5$ hours and each person hires their own racket, how many euros does each of the three original friends save when the fourth player joins?

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

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

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

To calculate the numerical value of the algebraic expression:

$$E = 7x^2y - 12x^2y + 8x^2y - 2x^2y$$

for $x = -\frac{2}{3}$ and $y = 9$, two students follow different paths:

• Method A (Cian): Directly substitutes the values of $x$ and $y$ into each of the four monomials separately, calculates the four numerical results, and then performs the additions and subtractions with the numbers obtained.

• Method B (Aoife): First combines like terms to reduce the entire expression to a single monomial, and then substitutes the numerical values only once.

1. a) Solve the exercise in detail following Cian's method and then following Aoife's.
2. b) Analyze both procedures: in which one did you have to do more operations with powers and fractions? Justify which of the two methods is more convenient and safer to avoid calculation errors.

A town hall designs two types of rectangular plots for community gardens:

• Type A garden has a base of $3x$ metres and a height of $2x^2y$ metres.

• Type B garden has a base of $k \times x$ metres (where $k$ is an unknown fixed number) and a height of $5x^2y$ metres.

1. a) Find the monomial that represents the area of each type of garden by multiplying its base by its height. State the coefficient, the literal part, and the degree of each.
2. b) Explain with reasoning why the two monomials obtained are like terms, whatever the value of the coefficient $k$.
3. c) The town hall wants to build $4$ type A gardens and $2$ type B gardens. Determine the value that the number $k$ must have so that the total area of gardens built is exactly equal to $44x^3y\text{ m}^2$. Show the entire deduction process by combining like terms.