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) $x + x + x + x$

Fully simplify the following algebraic expression by grouping like monomials:

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

Calculate the product of the following monomials. Remember to multiply the coefficients together and add the 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) $(-x^{2}) \times (-x^{3})$

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) $(-2z^3) \times (-5z^2) \times z$

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) $(4x^{2}) : (2x^{2})$

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

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) $(-2x)^{3}$

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

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

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

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

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 $

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:

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

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:

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

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.

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

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 $x$:

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 representing the total surface area of the frame (the shaded region).

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 $ meters. Express the final result in simplified form.

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 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).

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.