Cambridge Lower Secondary CheckpointStage 9

Algebra (9Ae)

Mathematics Stage 9 Chapter Notes

What this chapter covers

Algebra - Expressions, equations and formulae
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Algebra (9Ae) notes

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In this chapter

  1. 1Mastering the Laws of Indices
  2. 2Expanding Double Brackets
  3. 3Simplifying Algebraic Fractions
  4. 4Substitution into Expressions and Formulae
  5. 5Forming Expressions and Equations

1. Mastering the Laws of Indices

Indices (or powers) are a shorthand way of writing repeated multiplication of the same number. For example, instead of writing 5 × 5 × 5, we write 5³. The small '3' is the index, and the '5' is the base. There are several rules, called the laws of indices, that help us simplify expressions involving powers.

Multiplication Rule: aᵐ × aⁿ = aᵐ⁺ⁿ

Division Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power of a Power Rule: (aᵐ)ⁿ = aᵐⁿ

Zero Index Rule: a⁰ = 1 (for any non-zero 'a')

Power of One Rule: a¹ = a

Key term

Index (plural: Indices): A number that tells you how many times to multiply a base number by itself.

Examiner insight

Examiners look for correct application of the rules, especially when multiple rules are combined in one question. Show your intermediate steps clearly, such as dealing with numbers and each variable separately.

Common pitfall

A common mistake is to multiply the bases, for example, simplifying 3x² × 4x⁵ to 12x¹⁰ instead of the correct 12x⁷. Remember to add the powers but multiply the coefficients.

Worked example 11 mark

Simplify the expression: p⁴ × p⁵

  1. 1

    Identify the base, which is 'p' in both terms.

  2. 2

    The operation is multiplication, so we use the multiplication rule: aᵐ × aⁿ = aᵐ⁺ⁿ.

  3. 3

    Add the indices: 4 + 5 = 9.

  4. 4

    The simplified expression is p⁹.

Worked example 22 marks

Simplify fully: 15x⁹ ÷ 3x⁴

  1. 1

    First, divide the numerical coefficients: 15 ÷ 3 = 5.

  2. 2

    Next, simplify the algebraic part using the division rule: x⁹ ÷ x⁴.

  3. 3

    Subtract the indices: 9 - 4 = 5. This gives x⁵.

  4. 4

    Combine the numerical and algebraic parts.

  5. 5

    The final answer is 5x⁵.

Worked example 32 marks

Simplify the expression: (2y³)²

  1. 1

    This expression means (2y³) × (2y³).

  2. 2

    Alternatively, apply the power to everything inside the bracket.

  3. 3

    Apply the power to the number: 2² = 4.

  4. 4

    Apply the power to the variable using the 'power of a power' rule: (y³)² = y³ˣ² = y⁶.

  5. 5

    Combine the results: 4y⁶.

Recap

  • When multiplying terms with the same base, add the indices.
  • When dividing terms with the same base, subtract the indices.
  • When raising a power to another power, multiply the indices.
  • Anything to the power of zero is 1.
  • The laws of indices only apply to terms with the same base.

Quick check

  1. Simplify g⁶ × g⁷.1 mark
  2. Simplify 20h¹⁰ ÷ 5h².1 mark

2. Expanding Double Brackets

Expanding brackets means multiplying them out to remove them. When you have two brackets multiplied together, like (x + 2)(x + 3), you need to multiply every term in the first bracket by every term in the second bracket. A popular method to remember this is FOIL.

(a + b)(c + d) = ac + ad + bc + bd

FOIL Method: First, Outer, Inner, Last

Key term

Expand: To multiply out the terms within brackets to remove them from an expression.

Examiner insight

Marks are often awarded for correctly expanding to four terms before simplifying. Writing out the FOIL steps clearly helps prevent errors and can secure partial marks even if the final simplification is wrong.

Common pitfall

A very frequent error is seen with expressions like (x - 5)². Students often write x² - 25 or x² + 25, forgetting the middle term that comes from expanding (x - 5)(x - 5).

Fun fact

Expanding brackets is a key step in solving quadratic equations, which can model everything from the path of a thrown ball to the profit of a business.

Worked example 12 marks

Expand and simplify: (x + 4)(x + 6)

  1. 1

    Use the FOIL method.

  2. 2

    First: x × x = x²

  3. 3

    Outer: x × 6 = 6x

  4. 4

    Inner: 4 × x = 4x

  5. 5

    Last: 4 × 6 = 24

  6. 6

    Combine the terms: x² + 6x + 4x + 24

  7. 7

    Collect the like terms (the 'x' terms): 6x + 4x = 10x

  8. 8

    Final simplified answer: x² + 10x + 24

Worked example 22 marks

Expand and simplify: (y - 5)(2y + 3)

  1. 1

    Use the FOIL method, being careful with the negative sign.

  2. 2

    First: y × 2y = 2y²

  3. 3

    Outer: y × 3 = 3y

  4. 4

    Inner: -5 × 2y = -10y

  5. 5

    Last: -5 × 3 = -15

  6. 6

    Combine the terms: 2y² + 3y - 10y - 15

  7. 7

    Collect the like terms: 3y - 10y = -7y

  8. 8

    Final simplified answer: 2y² - 7y - 15

Worked example 32 marks

Expand and simplify: (a - 6)²

  1. 1

    First, write out what (a - 6)² means: (a - 6)(a - 6).

  2. 2

    Now use FOIL.

  3. 3

    First: a × a = a²

  4. 4

    Outer: a × -6 = -6a

  5. 5

    Inner: -6 × a = -6a

  6. 6

    Last: (-6) × (-6) = +36

  7. 7

    Combine the terms: a² - 6a - 6a + 36

  8. 8

    Collect the like terms: -6a - 6a = -12a

  9. 9

    Final simplified answer: a² - 12a + 36

Recap

  • FOIL stands for First, Outer, Inner, Last.
  • Every term in the first bracket multiplies every term in the second.
  • Be extremely careful with positive and negative signs during multiplication.
  • After expanding, always look for like terms to collect and simplify.
  • Remember that (x + a)² is not x² + a²; it is (x + a)(x + a).

Quick check

  1. Expand (x + 1)(x + 5).2 marks
  2. What is the first step to expanding (y - 4)²?1 mark

3. Simplifying Algebraic Fractions

Simplifying algebraic fractions is very similar to simplifying numerical fractions. The goal is to find common factors in the numerator (top) and the denominator (bottom) and 'cancel' them out. To do this, you may need to factorise the numerator or denominator first.

(ax)/(ay) = x/y

Key term

Factorise: To express an algebraic expression as a product of its factors, which often involves putting it into brackets.

Examiner insight

Examiners reward students who show clear factorisation of both the numerator and denominator before any cancellation is performed. This demonstrates a correct understanding of the process.

Common pitfall

Incorrectly 'cancelling' terms. For example, in (x + 4)/4, it is wrong to cancel the 4s to get 'x'. The numerator must be factorised first (if possible). Treat the numerator and denominator as if they are in invisible brackets.

Worked example 12 marks

Simplify the fraction: (12a³b) / (4ab)

  1. 1

    Look at the numbers first: 12 / 4 = 3.

  2. 2

    Look at the 'a' terms: a³ / a = a³⁻¹ = a².

  3. 3

    Look at the 'b' terms: b / b = b¹⁻¹ = b⁰ = 1.

  4. 4

    Combine the simplified parts: 3 × a² × 1.

  5. 5

    The final simplified fraction is 3a².

Worked example 22 marks

Simplify: (5x + 10) / 5

  1. 1

    The numerator has two terms. We cannot cancel the 5 from just one term.

  2. 2

    First, factorise the numerator by taking out the common factor, which is 5.

  3. 3

    5x + 10 = 5(x + 2).

  4. 4

    Now the fraction is: (5(x + 2)) / 5.

  5. 5

    The '5' on the top and bottom are now common factors and can be cancelled.

  6. 6

    The simplified expression is x + 2.

Recap

  • To simplify an algebraic fraction, cancel common factors from the numerator and denominator.
  • If the numerator or denominator contains a sum or difference, you must factorise it first.
  • You can only cancel factors that are multiplying the rest of the expression, not terms that are being added or subtracted.
  • Simplify the number part, then each letter part separately.

Quick check

  1. Simplify (8y⁶) / (2y³).1 mark
  2. What is the common factor of 6x + 9?1 mark

4. Substitution into Expressions and Formulae

Substitution is the process of replacing letters (variables) in an algebraic expression or formula with given numbers. Once the variables are replaced, you calculate the result. It is crucial to follow the correct order of operations, known as BIDMAS or BODMAS, to get the right answer.

BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction

Key term

Substitute: To replace a variable in an algebraic expression with a specific numerical value.

Examiner insight

Clear working is essential. Examiners want to see the substitution step written out before the final calculation. This can earn you method marks even if you make a later arithmetic slip.

Common pitfall

Making errors with negative numbers is the most common pitfall. For example, when finding x² with x = -3, many students calculate -3² = -9, but the correct answer is (-3)² = 9. Using brackets when substituting helps prevent this.

Worked example 12 marks

Find the value of the expression 4x - 7y when x = 5 and y = 2.

  1. 1

    Write down the expression: 4x - 7y.

  2. 2

    Substitute the given values: x = 5 and y = 2.

  3. 3

    The expression becomes: 4(5) - 7(2).

  4. 4

    Perform the multiplications first (BIDMAS): 20 - 14.

  5. 5

    Perform the subtraction: 20 - 14 = 6.

  6. 6

    The value is 6.

Worked example 22 marks

The formula for the area of a trapezium is A = ½(a + b)h. Find the area A when a = 6 cm, b = 10 cm and h = 5 cm.

  1. 1

    Write down the formula: A = ½(a + b)h.

  2. 2

    Substitute the values: A = ½(6 + 10) × 5.

  3. 3

    Work out the brackets first (BIDMAS): A = ½(16) × 5.

  4. 4

    Perform the multiplications: A = 8 × 5.

  5. 5

    The final answer is A = 40. Remember the units.

  6. 6

    A = 40 cm².

Worked example 33 marks

Find the value of 3p² - q when p = -4 and q = 8.

  1. 1

    Write down the expression: 3p² - q.

  2. 2

    Substitute the values, using brackets for the negative number: 3(-4)² - 8.

  3. 3

    Calculate the Index/Power first (BIDMAS): (-4)² = (-4) × (-4) = 16.

  4. 4

    The expression is now: 3(16) - 8.

  5. 5

    Perform the multiplication: 3 × 16 = 48.

  6. 6

    The expression is now: 48 - 8.

  7. 7

    Perform the subtraction: 48 - 8 = 40.

  8. 8

    The value is 40.

Recap

  • Always follow the order of operations (BIDMAS/BODMAS).
  • When substituting a negative number, always use brackets to avoid errors.
  • Write down the substitution step clearly before you start calculating.
  • Calculate powers and roots before multiplications and divisions.
  • Double-check your arithmetic, especially with negative numbers.

Quick check

  1. If a = 10 and b = 3, what is the value of a - 2b?1 mark
  2. What is the value of 5x² when x = -2?2 marks

5. Forming Expressions and Equations

In algebra, we use letters and symbols to represent unknown quantities and relationships. An 'expression' is a collection of terms representing a quantity (e.g., 2x + 3). An 'equation' states that two expressions are equal (e.g., 2x + 3 = 11). A key skill is translating a situation described in words into the language of algebra.

Key term

Equation: A mathematical statement that asserts the equality of two expressions, indicated by an equals sign (=).

Examiner insight

Examiners look for the ability to correctly translate each part of a word problem into its algebraic equivalent before combining them into a final expression or equation. Defining your variables is good practice.

Common pitfall

Confusing expressions with equations. An expression for '5 more than n' is 'n + 5'. An equation stating '5 more than n is 12' is 'n + 5 = 12'. Don't add '= 0' unless the question justifies it.

Fun fact

The word 'algebra' comes from the title of a book, 'Al-jabr wa'l-muqabala', written in the 9th century by the Persian mathematician Al-Khwarizmi. 'Al-jabr' referred to the process of moving a negative term to the other side of an equation.

Worked example 11 mark

A smartphone costs £P. A case for the phone costs £15. Write an expression for the total cost.

  1. 1

    Identify the cost of the phone: P.

  2. 2

    Identify the cost of the case: 15.

  3. 3

    The 'total cost' means we need to add these amounts together.

  4. 4

    The expression is P + 15.

Worked example 23 marks

A rectangle has a width of 'w' cm. Its length is 4 cm longer than its width. Write a simplified expression for its perimeter.

  1. 1

    Width = w.

  2. 2

    Length = w + 4 (since it's 4 cm longer).

  3. 3

    The perimeter is the distance around the shape: Width + Length + Width + Length.

  4. 4

    Perimeter = w + (w + 4) + w + (w + 4).

  5. 5

    Collect the like terms: (w + w + w +w) + (4 + 4).

  6. 6

    The simplified expression for the perimeter is 4w + 8.

Worked example 32 marks

Ben is 'x' years old. His sister, Chloe, is twice as old as him. The sum of their ages is 27. Write an equation to represent this information.

  1. 1

    Ben's age = x.

  2. 2

    Chloe is twice as old, so Chloe's age = 2x.

  3. 3

    The 'sum of their ages' means we add them: x + 2x.

  4. 4

    We are told this sum 'is' 27. The word 'is' translates to the equals sign.

  5. 5

    So, the equation is x + 2x = 27.

  6. 6

    This can be simplified to 3x = 27.

Recap

  • An expression represents a value and has no equals sign.
  • An equation connects two expressions with an equals sign.
  • Break down word problems into smaller parts and translate each part into algebra.
  • Look for keywords: 'sum' means add, 'product' means multiply, 'less than' means subtract, 'is' means equals.
  • Always define what your variable represents (e.g., let 'w' be the width).

Quick check

  1. A taxi costs £3 plus £2 per mile, m. Write an expression for the total cost.1 mark
  2. Is 5x - 10 an expression or an equation?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Simplify fully: a⁸ × a³1 mark
  2. Find the value of the expression 5p - 3q when p = 4 and q = -2.2 marks
  3. Expand and simplify: (x - 6)(x + 4)2 marks
  4. A rectangle has width w cm. Its length is 5 cm more than its width. Write a simplified expression for its area in cm².3 marks
  5. Simplify fully: (24c⁶d⁴) / (6c²d)2 marks
  6. Using the formula v = u + at, find the value of v when u = 15, a = -9.8 and t = 3.2 marks
  7. Expand and simplify: (2y - 5)²3 marks
  8. Find the expression that should replace the box: (m⁷ × □) / m⁴ = m⁹3 marks
  9. The area of a rectangle is given as (x² + 7x + 12) cm². The width of the rectangle is (x + 3) cm. Find an expression for the length of the rectangle.3 marks
  10. Tickets for a concert cost £a for an adult and £c for a child. Write an equation to show that the total cost for 3 adults and 5 children is £155.2 marks

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