Exercises: Operations with monomials

To understand how mathematical expressions are constructed, it is essential to know the parts of a monomial. Complete the requested information for each of the following monomials:

1. a) $7x^{3}$
2. b) $-a^{2}b$
3. c) $\frac{1}{2}x^{4}y^{2}$
4. d) $-5$

For each one, state:

• The coefficient.

• The literal part.

• The degree of the monomial.

In algebra, we can only add or subtract "like terms". Identify which of the following monomials are like terms and group them:

$5x^{2}$, $-3x$, $\frac{1}{4}x^{2}$, $7y$, $x$, $-x^{2}$, $12x$, $2y$

Simplify the following expressions by adding and subtracting like monomials:

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

Calculate the product of the following monomials. Remember to multiply the coefficients together and add the exponents of identical letters:

1. a) $(3x^{2}) \times (5x^{4})$
2. b) $(-2a^{3}) \times (4a)$
3. c) $(\frac{1}{2}x) \times (6x^{2})$
4. d) $(-x^{2}) \times (-x^{3})$

Perform the following divisions of monomials. Remember to divide the coefficients and subtract the exponents of the letters:

1. a) $(12x^{5}) : (3x^{2})$
2. b) $(-10a^{4}) : (5a^{3})$
3. c) $(15x^{3}y^{2}) : (3x^{2}y)$
4. d) $(4x^{2}) : (2x^{2})$

Calculate the power of these monomials. Do not forget to raise both the coefficient and the literal part to the indicated exponent:

1. a) $(2x^{3})^{2}$
2. b) $(-3a^{2})^{2}$
3. c) $(x^{4})^{3}$
4. d) $(-2x)^{3}$

Find the algebraic expression representing the perimeter of the following rectangular plot for a school garden. Once you have the expression, fully simplify it by combining like terms.

In a racing video game, for each gold coin collected you get $5x$ points and for each silver coin you get $2x$ points.

If in a game a player has collected 4 gold coins and 6 silver coins:

1. a) Write the monomial representing the total points from the gold coins.
2. b) Write the monomial representing the total points from the silver coins.
3. c) Add both monomials to find the total points for the game.

You can do it on paper and send a photo with your answer.

In Mathematics class, Cian solved the following power of a monomial on the board:

$$(-3x^4)^2 = -6x^8$$

His classmate Aoife raises her hand and claims that Cian has made two different conceptual errors in that single step.

1. a) Identify the two errors made by Cian, clearly explaining which rule regarding operations with monomials he applied incorrectly in each case (one in the coefficient and another in the sign or in the exponents).
2. b) Write the correct, fully expanded step-by-step resolution, justifying the final sign, coefficient, and exponent.

A teacher partially erased the blackboard at the end of class, leaving these three operations with monomials incomplete:

1. a) $4x^3 \times \Box = -20x^7$
2. b) $\frac{\Box}{3x^2} = 5x^4$
3. c) $(\Box)^3 = -8x^6$

Find out which monomial must be placed in each box ($\Box$) to make the equalities true. In each case, explain with reasoning how you deduce the sign, the numerical coefficient, and the exponent of the literal part using the corresponding inverse operation.

Fully simplify the following algebraic expression by grouping like monomials:

$9x^3 - 5x^2 + 4x - 2x^3 + 7x^2 - x + 10$

Calculate the result of the following monomial multiplications and simplify the literal part using index laws:

1. a) $(4a^2b) \times (-3ab^3)$

1. b) $(\frac{1}{2}x^4) \times (6x^2)$

1. c) $(-2z^3) \times (-5z^2) \times z$

Find the simplified algebraic expression representing the perimeter of the following plot designated for a play area in a park:

Work out the following powers of monomials. Remember that the exponent affects both the coefficient and each of the variables in the literal part:

1. a) $(-3x^4)^2$

1. b) $(2a^2b^3)^4$

1. c) $(-\frac{1}{2}m^5)^3$

Simplify the following monomial quotients. Express the result as a single monomial where possible:

1. a) $\frac{20x^6}{5x^2}$

1. b) $(-12a^3b^2) : (3a^2b^2)$

1. c) $\frac{15m^4n^2}{10m^2}$

In a tree-planting campaign, each volunteer plants $3x$ trees in the morning and $2x$ trees in the afternoon. If there are a total of $y$ volunteers, write the simplified algebraic expression representing:

1. a) The total number of trees planted by a single volunteer in one day.

1. b) The total number of trees planted by all volunteers in a week (7 days).

Find the expression representing the area of the shaded region in the following figure, knowing that the side of the outer square measures $4x$ and the side of the inner white square measures $x$:

Solve the following combined operations with monomials, respecting the order of operations:

1. a) $5x^2 \times 2x + 12x^4 : 3x$

1. b) $3ab \times (2a^2b) - (5a^3b^2)$

1. c) $(2n^3)^2 - 5n^6 + n^2 \times n^4$

Consider the following combined operation with monomials:

$$M = \frac{(-2x^3)^3 \times (6x^2)}{-4x^5}$$

1. a) Before performing any numerical calculation with the coefficients, make a reasoned estimation by answering these two questions:

1. What sign will the final result have? Justify it by successively analyzing the signs of the power, of the product in the numerator, and of the final division.

2. What will be the exact degree of the resulting monomial? Justify it by mentally calculating the addition and subtraction of exponents according to the properties of powers.

1. b) Solve the combined operation step by step respecting the order of operations and simplify the resulting monomial.
2. c) Check whether the result you obtained matches your previous deductions about the sign and degree in section a).

An agricultural cooperative in Tomelloso (Castilla-La Mancha) installs solar panels on a rectangular plot to power irrigation pumps. The dimensions of the land are expressed in terms of a variable $x$ (in meters): the width measures $3x$ meters and the length is three times the width.

1. a) Determine the simplified monomial representing the total area of the plot in square meters.
2. b) To distribute the wiring, they divide the entire plot into identical square subplots with a side length of $x$ meters. Write the monomial that represents the area of one of these subplots and calculate how many subplots fit in total by carrying out the appropriate division of monomials.
3. c) The annual maintenance cost of the plot is $0.5$ euros per square meter of surface area. If the cooperative receives a fixed annual regional subsidy of $15x^2$ euros, determine through operations with monomials whether the subsidy is sufficient to cover maintenance and how much money is left over or short in terms of $x$.

Work out the following combined operation with monomials, fully simplifying the result. Pay special attention to the order of operations and power rules:

$$ \frac{2}{3}x^2 \times (-3x^3)^2 - 5x^5 \times (2x^3) + \frac{16x^{10}}{2x^2} $$

A design company manufactures aluminium frames for tablet screens. The design follows the proportions shown in the following figure, where all measurements are expressed in terms of a variable $x$. Find the monomial representing the total surface area of the frame (the shaded region).

Find the unknown monomials $M$ and $N$ that make the following equalities true:

1. a) $ (-4x^3y^2) \times M = 20x^7y^5 $

1. b) $ \frac{N}{-3ab} = 6a^2b^3 $

1. c) $ (2x^2)^3 : M = -4x $

In a strategy video game, a battalion's energy is calculated by multiplying its attack level by its defence level, divided by its fatigue. Given that:

• Attack: $ 12a^2b^3 $

• Defence: $ 5a^3b $

• Fatigue: $ 10ab^2 $

Express the total energy of the battalion as a single monomial and determine its degree.

Factorise out the highest common factor in the following algebraic expression. Remember to extract both the highest common factor of the coefficients and the common variables with their lowest exponent:

$$ 24x^5y^4z^2 - 36x^3y^6z^3 + 12x^2y^4z^5 $$

Calculate the value of the resulting monomial at the top of this "product pyramid", where each brick is the product of multiplying the two monomials directly below it:

Simplify the following expression by applying the distributive property and combining the resulting like monomials:

$$ -3ab^2 \times (2a^2 - 5ab + b^2) - 2a^2b \times (4ab - a) $$

Calculate the monomial representing the volume of a 3D shape formed by joining 5 identical cubes, where the edge of each cube measures $ \frac{2}{3}x^2 $ metres. Express the final result in simplified form.

To calculate the numerical value of the algebraic expression:

$$E = \frac{(6x^4)^2 \times x^3}{12x^7}$$

for the value $x = -2$, two 2nd Year students propose two different strategies:

• Seán's method: Substitutes the value $x = -2$ directly into each $x$ of the original expression, calculates the resulting numerical powers, and finally carries out the multiplications and division with the obtained numbers.

• Aoife's method: First applies the properties of operations with monomials to simplify the expression into a single irreducible monomial, and only at the end substitutes $x = -2$.

1. a) Apply Seán's method, indicating the intermediate numbers he obtains in the numerator and denominator before simplifying.
2. b) Apply Aoife's method, first simplifying the algebraic expression and then calculating the final numerical value.
3. c) Compare both methods and justify which of the two is more advantageous and safer in mathematics. Support your answer by assessing the size of the numbers handled and the likelihood of making operational errors in each procedure.

A metal hopper for storing grain is to be designed, with a capacity given by the volume of a rectangular cuboid. The total volume is the product of three monomials corresponding to its three dimensions (length $L(x)$, width $A(x)$, and height $H(x)$), such that $V(x) = L(x) \times A(x) \times H(x)$. The following technical data are known:

• The monomial for the total volume is $V(x) = 72x^7$.

• The height of the hopper is the monomial $H(x) = 2x^2$.

• Both the length $L(x)$ and the width $A(x)$ have positive integer coefficients.

• The degree of the length $L(x)$ is one unit greater than the degree of the width $A(x)$.

• The numerical coefficient of the length $L(x)$ is exactly four times the coefficient of the width $A(x)$.

1. a) Deduce step by step the exact monomials that correspond to $L(x)$ and $A(x)$, justifying how you obtain both the coefficients and the degrees from the properties of operations with monomials.
2. b) State the expression for the area of the base of the hopper, $B(x) = L(x) \times A(x)$. Next, answer with justification: is it possible to modify the dimensions $L(x)$ and $A(x)$ (keeping their product the same and ensuring they remain monomials) so that the degree of the base is 6? Justify why it is or is not possible based on the relationship between the degree of the volume and that of the height.