Operations with Monomials

To understand how mathematical expressions are built, 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 the world of algebra, we can only add or subtract "like terms". Identify which of the following monomials are like terms and group them together:

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

Simplify the following expressions by adding and subtracting the like terms:

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

Reduce the following algebraic expression as much as possible by grouping the like terms:

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

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

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

Calculate the result of the following multiplications of monomials and simplify the literal part using the properties of powers:

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

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

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

Work out the following divisions of monomials. Remember to divide the coefficients and subtract the indices 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) $(4x^{2}) : (2x^{2})$

Simplify the following quotients of monomials. Express the result as a single monomial when possible:

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

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

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

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

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

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

Work out the following combined operations with monomials, respecting the order of operations:

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

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

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

Perform the following combined operation with monomials, simplifying the result as much as possible. Pay special attention to the order of operations and the rules of powers:

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

Reduce the following expression by applying the distributive property and simplifying the resulting like terms:

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

Extract the greatest possible common factor in the following algebraic expression. Remember that you must extract both the greatest common factor of the coefficients and the common variables with their lowest exponent:

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

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

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

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

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

In a strategy video game, the energy of a battalion is calculated by the product of its attack level and its defense level, divided by its fatigue. If we know that:

• Attack: $ 12a^2b^3 $

• Defense: $ 5a^3b $

• Fatigue: $ 10ab^2 $

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

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

Find the algebraic expression that represents the perimeter of the following rectangular plot intended for a school garden. Once you have the expression, simplify it fully by adding its like terms.

Find the simplified algebraic expression that represents the perimeter of the following plot intended for a play area in a park:

Calculate the expression that represents 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 $x$:

A design company manufactures aluminum 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 that represents the total surface area of the frame (the shaded zone).

In a racing video game, for every gold coin collected you get $5x$ points and for every 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 that represents the total points from the gold coins.
2. b) Write the monomial that represents the total points from the silver coins.
3. c) Add both monomials to find the total points for the game.

In a reforestation campaign, each volunteer plants $3x$ trees in the morning and $2x$ trees in the afternoon. If there is a total of $y$ volunteers, write the simplified algebraic expression that represents:

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 the volunteers in a week (7 days).

Calculate the monomial that represents the volume of a geometric solid formed by joining 5 identical cubes, where the edge of each cube measures $ \frac{2}{3}x^2 $ feet. Express the final result in a simplified form.