Algebraic notation and simplifying expressions

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Apuntes de la lección

Algebraic Notation

When a letter and a number are written together, they are multiplied: 3y means 3 × y.
When two letters are written together, they are multiplied: ab means a × b.
A power shows repeated multiplication: a² means a × a, and a³ means a × a × a.
Division is written as a fraction: a/b means a ÷ b.
The multiplication sign is usually left out in algebra to avoid confusion with the letter x.

Key Words in Algebra

A term is a single number, variable, or product of numbers and variables, such as 5, 3x, or 2ab.
An expression is a collection of terms combined by + or −, such as 3x + 2y − 5.
An equation states that two expressions are equal, such as 3x + 2 = 11.
A formula is a rule connecting variables, such as A = l × w (A = area, l = length, w = width).
An identity is true for all values of the variables, such as 2(x + 3) = 2x + 6.

Like Terms

Like terms are terms that have exactly the same variables raised to the same powers.
For example, 5x and 3x are like terms because both contain x to the power 1.
The numbers multiplying the variables are called coefficients; in 5x, the coefficient is 5.
Unlike terms, such as 2x and 3y, cannot be combined because the variables are different.

Collecting Like Terms

To collect like terms, add or subtract their coefficients and keep the common variable part.
For example: 5x + 3x = (5 + 3)x = 8x.
Similarly, 7y − 2y = (7 − 2)y = 5y.
Only like terms can be collected; for example, 4x + 3y cannot be simplified further.
Collecting like terms is an important step when solving equations.

Simplifying Products

When multiplying terms, multiply the coefficients and multiply the variables.
For example: 3a × 4b = (3 × 4) × (a × b) = 12ab.
Use index laws: when multiplying powers of the same variable, add the indices, e.g. x² × x³ = x²⁺³ = x⁵.
The order of multiplication does not matter, so 2 × x × y can be written as 2xy.

Simplifying Quotients

When dividing terms, divide the coefficients and divide the variables.
For example: 12x ÷ 4 = 3x.
Use index laws: when dividing powers of the same variable, subtract the indices, e.g. x⁵ ÷ x² = x⁵⁻² = x³.
Any variable divided by itself equals 1, e.g. y ÷ y = 1.

Index Laws in Algebra

Multiplication law: a^m × aⁿ = a^m+n.
Division law: a^m ÷ aⁿ = a^m−n.
Power of a power law: (a^m)ⁿ = a^m×n.
Zero index law: a⁰ = 1 (for any non-zero a).
These laws work for any variable or number base.

Simplifying Expressions with Brackets

To simplify an expression with brackets, first expand the brackets by multiplying each term inside by the number or variable outside.
For example: 3(4x²y − 6y) = 12x²y − 18y.
After expanding, collect any like terms.
For example: 3(4x²y − 6y) + 7x²y − 3y² + 2(8y − 4y² − 4x²y) simplifies to 11x²y − 11y² − 2y.

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Preguntas de práctica

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1.In algebra, how is the product a × b normally written?
Easy
- Aab
- Ba + b
- Ca.b
- Da × b
2.What does the expression a² mean?
Easy
- Aa × a
- Ba + a
- C2 × a
- Da ÷ a
3.Simplify: 5x + 3x
Easy
- A8x
- B15x
- C8x²
- D2x
4.Simplify: 4a + 3b − 2a + 5b
Medium
- A2a + 8b
- B2a + 2b
- C6a + 8b
- D2a − 8b
5.Which of the following are like terms? (Select all that apply)
Medium
- A3x²y
- B−5x²y
- C7xy²
- D2x²y
- E4y²x
6.An expression is a mathematical statement that two things are equal.
Easy
True or false?
7.Match each algebraic expression with its simplified form.
Easy
- 5x + 3x
- 7y − 4y
- 2a × 3b
- x² × x³
- 8x
- 3y
- 6ab
- x⁵
8.Arrange the steps for simplifying 3(4x²y − 6y) + 7x²y − 3y² + 2(8y − 4y² − 4x²y) in the correct order.
Medium
- Expand the brackets: 12x²y − 18y + 7x²y − 3y² + 16y − 8y² − 8x²y
- Group like terms: (12x²y + 7x²y − 8x²y) + (−3y² − 8y²) + (−18y + 16y)
- Combine coefficients: 11x²y − 11y² − 2y

Más preguntas

9.Which expression represents the perimeter P?
Medium
- A6x + 8
- B3x + 4
- C6x + 4
- D2x + 8
10.Arrange the steps for simplifying 2(3x + 4) − 5x in the correct order.
Medium
- Expand the brackets: 6x + 8 − 5x
- Collect like terms: (6x − 5x) + 8
- Simplify: x + 8
11.Simplify: 4m + 7n − 2m + 3n
Easy
- A2m + 10n
- B6m + 4n
- C2m + 4n
- D6m + 10n
12.Simplify: 3(2x + 5) − 4x
Medium
- A2x + 15
- B6x + 15
- C2x + 5
- D6x − 15
13.Which of the following expressions are equivalent to 2(3a + 4b) − 3a? (Select all that apply)
Medium
- A6a + 8b − 3a
- B3a + 8b
- C6a + 4b − 3a
- D3a + 4b
- E8b + 3a
14.The terms 4x² and 4x are like terms.
Easy
True or false?
15.Simplify: 2(3x² + 4x) − 3(x² − 2x)
Medium
- A3x² + 14x
- B3x² + 2x
- C9x² + 2x
- D3x² − 14x
16.Simplify: 7x + 3y + 2x − y
Easy
- A9x + 2y
- B9x + 4y
- C10x + 2y
- D9x − 2y
17.Simplify: 5(2a + 3) − 4a
Medium
- A6a + 15
- B10a + 15 − 4a
- C6a + 3
- D10a + 15
18.Simplify: 3 × 5y
Easy
- A15y
- B8y
- C15 + y
- D15y²
19.Simplify: 12x² ÷ 4x
Medium
- A3x
- B3x²
- C8x
- D3
20.Simplify: 2(3x + 4y) − (x − 2y)
Medium
- A5x + 10y
- B5x + 6y
- C5x + 2y
- D6x + 8y − x − 2y
21.Which of the following expressions are equivalent to 4(2a − b) + 3b? (Select all that apply)
Medium
- A8a + b
- B8a − b
- C8a − 4b + 3b
- D8a − b + 3b
22.The terms 3x² and 5x² are like terms.
Easy
True or false?
23.Match each expression with its simplified form.
Medium
- 2x + 3x
- 4a − a
- 5y × 2
- 10b ÷ 2
- 5x
- 3a
- 10y
- 5b
24.A rectangle has length 3x + 4 and width 2x. Which expression represents its perimeter?
Medium
- A10x + 8
- B6x + 8
- C10x + 4
- D5x + 4
25.A triangle has side lengths 2a + 3, 4a − 1 and a + 5. Its perimeter is 47. What is the value of a?
Hard
- A5
- B6
- C7
- D4
26.A shop sells pens at p pence each and pencils at q pence each. Amira buys 4 pens and 3 pencils. Ben buys 2 pens and 5 pencils. Which expression gives the total cost in pence of their combined purchases?
Medium
- A6p + 8q
- B6p + 2q
- C8p + 6q
- D4p + 8q
27.A rectangle has length 2x + 5 and width x − 3. Which of the following statements are correct? (Select all that apply)
Hard
- AThe perimeter is 6x + 4
- BThe perimeter is 3x + 2
- CThe perimeter is 2(3x + 2)
- DThe perimeter is 6x − 4
- EThe perimeter is 6x + 16
28.In the expression 3a + 5b, the terms 3a and 5b are like terms.
Medium
True or false?
29.Match each contextual situation with the correct simplified expression for the total.
Medium
- 3 boxes each holding 2x + 5 items, plus 4 loose items
- A rectangle with sides 3x and x + 2, giving its perimeter
- 5 identical bags of 2x − 1 items
- 2(3x + 4) − x
- 6x + 19
- 8x + 4
- 10x − 5
- 5x + 8
30.Arrange the steps for finding the perimeter of a rectangle of length 3x + 2 and width 2x − 1, in the correct order.
Medium
- Write perimeter = 2(3x + 2 + 2x − 1)
- Collect like terms inside the bracket to get 5x + 1
- Multiply by 2 to get 10x + 2
- State the simplified perimeter 10x + 2
31.A rectangle has length 3x + 1 and width x + 4. A square has side length 2x. The perimeter of the rectangle is equal to the perimeter of the square. Find x.
Hard
- A2.5
- B3
- C1.5
- D4
32.Which expression gives the perimeter of the garden in metres?
Medium
- A16x
- B8x
- C15x
- D16x²
33.A sequence of shapes is made from matchsticks. Shape 1 uses 4 sticks, shape 2 uses 7 sticks, shape 3 uses 10 sticks. Each new shape adds 3 sticks. Which expression gives the number of sticks in shape n?
Challenge
- A3n + 1
- B3n + 4
- C4n
- Dn + 3
34.In the term 7x, what is the number 7 called?
Easy
- AThe coefficient
- BThe variable
- CThe index
- DThe constant
35.Which of these is the usual way to write a ÷ b in algebra?
Easy
- A\frac{a}{b}
- Bab
- Ca + b
- Da - b
36.Which of these expressions is a formula?
Easy
- AA = l × w
- B3x + 2
- C2x = 8
- D5 + 3 = 8
37.Which of these is an identity?
Medium
- A2(x + 3) = 2x + 6
- B2x + 3 = 7
- Cx + 1
- D3x = 12
38.Simplify: a × a × a
Easy
- Aa³
- B3a
- Ca²
- Da + 3
39.Simplify: 4x × 3x
Medium
- A12x²
- B12x
- C7x²
- D7x
40.Simplify: 10y³ ÷ 2y
Medium
- A5y²
- B5y³
- C8y²
- D5y
41.What is the numerical factor of a term called?
Easy
42.Simplify: 3 × 4y
Medium
43.Simplify the expression by collecting like terms: 3(4x²y − 6y) + 7x²y − 3y² + 2(8y − 4y² − 4x²y)
Medium
The simplified expression is ____.
44.Find the perimeter when x = 5.
Medium
45.Simplify by collecting like terms: 5p + 3q − 2p + 4q = ___ p + ___ q
Medium
5p + 3q − 2p + 4q = ____ p + ____ q
46.Simplify: 5x + 2y − 3x + 4y. What is the coefficient of x in the simplified expression?
Medium
47.What is the number in front of a variable in a term called?
Easy
48.Simplify the expression 3(2x + 4) − 2(x − 1). What is the coefficient of x in the simplified expression?
Medium
49.Simplify by collecting like terms: 7a + 2b − 3a + 5b = ___ a + ___ b
Easy
7a + 2b − 3a + 5b = ____ a + ____ b
50.Simplify 3 × 7y. What is the coefficient in your simplified expression?
Medium
51.Simplify 15x³ ÷ 3x. What is the power of x in the simplified expression?
Medium
52.Complete the statement about algebraic notation.
Easy
The expression a × b is usually written as ____, and a ÷ b is usually written as ____.
53.Explain the difference between an expression and an equation, giving an example of each.
Medium
54.Explain how to simplify the expression 4(2x + 3y) − 2(3x − y).
Medium
55.Explain how to simplify the expression 3(2x + 5) − 4(x − 2).
Hard
56.Explain the difference between a formula and an identity, giving an example of each.
Medium
57.Explain how to simplify the expression 5(2x + 3) − 2(x − 4).
Hard
58.Simplify: 3 × 4y
Medium
- A14 y
- B12 y
- C11 y
- D13 y
59.Find the perimeter when x = 5.
Medium
- A38
- B34
- C42
- D46
60.Simplify: 5x + 2y − 3x + 4y. What is the coefficient of x in the simplified expression?
Medium
- A2
- B1
- C3
- D4
61.Simplify the expression 3(2x + 4) − 2(x − 1). What is the coefficient of x in the simplified expression?
Medium
- A5
- B6
- C4
- D3
62.Simplify 3 × 7y. What is the coefficient in your simplified expression?
Medium
- A19
- B23
- C21
- D25
63.Simplify 15x³ ÷ 3x. What is the power of x in the simplified expression?
Medium
- A2
- B1
- C4
- D3

Apuntes de repaso

## Algebraic notation and simplifying expressions ### Notation conventions - **ab** means a × b; **3y** means 3 × y; **a²** means a × a; **a/b** means a ÷ b. - A **term** is a single number, variable, or product/quotient of these, e.g. 5x, 3y², 7. - An **expression** is a sum of terms, e.g. 5x + 3x. - An **equation** states two expressions are equal, e.g. 5x + 3x = 8x. - A **formula** relates quantities, e.g. P = F / A (P = pressure, F = force, A = area). - An **identity** is true for all values, e.g. 5x + 3x ≡ 8x. ### Like terms Like terms contain the same variables raised to the same powers; they may differ only by a numerical factor (the coefficient). - Example: 5x and 3x are like terms; 5x and 3y are not. - Combine like terms by adding or subtracting their coefficients and keeping the common variable part: - 5x + 3x = (5 + 3)x = 8x - ax + bx = (a + b)x ### Simplifying expressions 1. Remove brackets by multiplying each term inside by the factor outside. 2. Identify like terms (same variables and powers). 3. Collect like terms by combining coefficients. **Worked example** Simplify: 3(4x²y − 6y) + 7x²y − 3y² + 2(8y − 4y² − 4x²y) Step 1 — expand brackets: 12x²y − 18y + 7x²y − 3y² + 16y − 8y² − 8x²y Step 2 — group like terms: - x²y terms: 12x²y + 7x²y − 8x²y = 11x²y - y² terms: −3y² − 8y² = −11y² - y terms: −18y + 16y = −2y Result: 11x²y − 11y² − 2y ### Index laws in algebra - a^m × a^n = a^{m+n} - a^m ÷ a^n = a^{m−n} - (a^m)^n = a^{mn} - a^0 = 1 (for a ≠ 0) - a^{−n} = 1 / a^n ### Key points to remember - Only like terms can be collected. - Coefficients are the numerical factors of terms. - Always simplify by expanding brackets first, then collecting like terms.

Term: A single number, variable, or product/quotient of these, e.g. 5x, 3y², 7.
Expression: A sum of terms, e.g. 5x + 3x.
Equation: A statement that two expressions are equal, e.g. 5x + 3x = 8x.
Formula: A rule relating quantities, e.g. P = F / A (P = pressure, F = force, A = area).
Identity: An equation true for all values of the variables, e.g. 5x + 3x ≡ 8x.
Like terms: Terms that contain the same variables raised to the same powers, possibly with different coefficients, e.g. 5x and 3x.
Coefficient: The numerical factor of a term, e.g. in 5x the coefficient is 5.
Collecting like terms: Combining like terms by adding or subtracting their coefficients and keeping the common variable part, e.g. 5x + 3x = 8x.

Tarjetas

How do you write a × b using algebraic notation?

ab (the multiplication sign is left out).

How do you write 3 × y using algebraic notation?

3y (the number is written before the letter, and the multiplication sign is left out).

How do you write a × a using algebraic notation?

a² (read as 'a squared').

How do you write a ÷ b using algebraic notation?

a/b or \frac{a}{b}.

What is a term in algebra?

A term is a part of an expression separated by + or − signs. It can be a number, a variable, or a product of numbers and variables.

What is an expression in algebra?

An expression is a mathematical phrase made from terms combined with + or − signs, for example 3x + 2y − 5.

What is an equation?

An equation is a statement that two expressions are equal, for example 3x + 2 = 11. It contains an equals sign.

What is a formula?

A formula is a rule or relationship written using algebra, for example A = l × w (area of a rectangle).

What is an identity?

An identity is an equation that is true for all values of the variables, for example 2(x + 3) = 2x + 6.

What are like terms?

Like terms are terms that have exactly the same variables raised to the same powers. They may have different coefficients.

How do you collect like terms?

Add or subtract the coefficients of the like terms and keep the common variable part the same. For example, 5x + 3x = 8x.

Simplify 5x + 3x.

Simplify 7a − 2a.

Simplify 4x² + 3x².

7x²

Simplify 2a + 3b + 4a.

6a + 3b (2a and 4a are like terms; 3b is not like them).

Simplify 3(2x + 5).

6x + 15

Simplify 2x × 3x.

6x²

Simplify 12x³ ÷ 4x.

3x²

What is the coefficient in the term 7y?

What is the coefficient in the term −3ab?

−3

Simplify x² × x³.

x⁵ (add the indices: 2 + 3 = 5).

Simplify y⁵ ÷ y².

y³ (subtract the indices: 5 − 2 = 3).

Simplify (x²)³.

x⁶ (multiply the indices: 2 × 3 = 6).

Simplify 3(4x²y − 6y) + 7x²y − 3y² + 2(8y − 4y² − 4x²y).

11x²y − 11y² − 2y. (Expand brackets first: 12x²y − 18y + 7x²y − 3y² + 16y − 8y² − 8x²y, then collect like terms.)

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