Cambridge Lower Secondary CheckpointStage 9

Number (9Ni)

Mathematics Stage 9 Chapter Notes

What this chapter covers

Number - Integers, powers and roots
ShareWhatsAppPost
Number (9Ni) notes

Unable to load PDF

The notes viewer could not load. Please refresh the page.

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Number (9Ni) notes as text: skim, search, and jump between subtopics.

~7 min read

In this chapter

  1. 1Mastering Indices: Negative and Zero Powers
  2. 2Rational vs. Irrational Numbers
  3. 3Estimating Roots and Surds

1. Mastering Indices: Negative and Zero Powers

Indices, or powers, tell us how many times to multiply a number by itself. But what about negative or zero indices? A negative index means we take the reciprocal of the number with a positive index. For example, 5⁻² is the same as 1/5². A zero index is even simpler: any non-zero number raised to the power of zero is always 1. The standard laws of indices for multiplication and division still apply to negative and zero indices.

a⁻ⁿ = 1/aⁿ

a⁰ = 1 (for a ≠ 0)

aᵐ × aⁿ = aᵐ⁺ⁿ

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Key term

Reciprocal: The reciprocal of a number is 1 divided by that number; for example, the reciprocal of x is 1/x.

Examiner insight

Examiners look for a clear understanding of the index laws, especially when combining them in multi-step problems. Show your working by writing out the intermediate steps, like aᵐ⁺ⁿ.

Common pitfall

A common mistake is thinking a⁻ⁿ means -aⁿ. Remember, the negative sign in the index indicates a reciprocal, not a negative result.

Worked example 12 marks

Calculate the value of 3⁻⁴.

  1. 1

    The negative index means we take the reciprocal.

  2. 2

    3⁻⁴ = 1 / 3⁴

  3. 3

    Calculate 3⁴: 3 × 3 × 3 × 3 = 81.

  4. 4

    So, 3⁻⁴ = 1/81.

Worked example 22 marks

Simplify 8³ × 8⁻⁵, leaving your answer as a single power of 8.

  1. 1

    The bases are the same (8), so we can use the multiplication law for indices: aᵐ × aⁿ = aᵐ⁺ⁿ.

  2. 2

    Add the indices: 3 + (-5) = 3 - 5 = -2.

  3. 3

    So, 8³ × 8⁻⁵ = 8⁻².

Worked example 33 marks

Simplify (6⁻²) ÷ (6⁻⁶). Give your answer as a whole number.

  1. 1

    The bases are the same (6), so we use the division law for indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

  2. 2

    Subtract the indices: (-2) - (-6) = -2 + 6 = 4.

  3. 3

    So, (6⁻²) ÷ (6⁻⁶) = 6⁴.

  4. 4

    Calculate the final value: 6⁴ = 6 × 6 × 6 × 6 = 1296.

Recap

  • A negative index means 'one over' the positive index, e.g., x⁻² = 1/x².
  • Any non-zero number to the power of 0 is 1, e.g., 99⁰ = 1.
  • When multiplying powers with the same base, add the indices.
  • When dividing powers with the same base, subtract the second index from the first.
  • Always simplify the indices first before calculating the final value.

Quick check

  1. What is the value of 10⁻³ as a decimal?1 mark
  2. Simplify y⁻⁷ ÷ y⁻³ as a single power.1 mark

2. Rational vs. Irrational Numbers

All numbers you encounter can be classified as either rational or irrational. A rational number is any number that can be written as a fraction p/q, where p and q are integers and q is not zero. This group includes all integers (e.g., 5 = 5/1), all terminating decimals (e.g., 0.75 = 3/4), and all recurring decimals (e.g., 0.333... = 1/3). Irrational numbers are the opposite: they cannot be expressed as a simple fraction. Their decimal representations go on forever without repeating. Famous examples include π and surds.

Key term

Surd: A surd is an irrational number that is the root (square root, cube root, etc.) of an integer that is not a perfect square, cube, etc.

Common pitfall

Students often assume all square roots are irrational. Remember that the square root of a perfect square (like √49 = 7) is a rational integer.

Fun fact

The decimal representation of π has been calculated to over 100 trillion digits, and it never repeats or ends, a key characteristic of irrational numbers.

Worked example 14 marks

Sort the following numbers into two groups: Rational and Irrational. √36, √37, -8, π, 4.5, 1/3.

  1. 1

    Analyse each number:

  2. 2

    √36 = 6. This is an integer, so it's rational.

  3. 3

    √37 cannot be simplified to an integer. It is a surd, so it's irrational.

  4. 4

    -8 can be written as -8/1. It's an integer, so it's rational.

  5. 5

    π is a famous irrational number.

  6. 6

    4.5 can be written as 9/2. It's a terminating decimal, so it's rational.

  7. 7

    1/3 is already a fraction, so it's rational.

  8. 8

    Rational Group: √36, -8, 4.5, 1/3.

  9. 9

    Irrational Group: √37, π.

Worked example 23 marks

Explain why √16 is rational but ∛16 is irrational.

  1. 1

    Consider √16. The square root of 16 is 4.

  2. 2

    Since 4 is an integer, it can be written as a fraction (4/1), so √16 is rational.

  3. 3

    Consider ∛16. We are looking for a number that, when cubed, gives 16.

  4. 4

    2³ = 8 and 3³ = 27. There is no integer that can be cubed to make 16.

  5. 5

    Therefore, ∛16 is a surd, which means it is an irrational number.

Recap

  • Rational numbers can be written as a fraction (p/q).
  • Integers, terminating decimals, and recurring decimals are all rational.
  • Irrational numbers cannot be written as a fraction.
  • The decimal form of an irrational number is non-terminating and non-recurring.
  • Surds, like √2 and ∛10, are irrational roots.
  • The root of a perfect square/cube (e.g., √81 = 9) is rational.

Quick check

  1. Is the number 0.272727... rational or irrational? Explain your answer.2 marks

3. Estimating Roots and Surds

Without a calculator, you can find a good estimate for an irrational root (a surd) by finding the two integers it lies between. To do this, you 'sandwich' the number inside the root between two perfect squares (for square roots) or perfect cubes (for cube roots). For example, to estimate √55, you know that 55 is between the perfect squares 49 (which is 7²) and 64 (which is 8²). Therefore, √55 must be between 7 and 8.

Key term

Consecutive Integers: Consecutive integers are whole numbers that follow each other in order, such as 4 and 5, or -2 and -1.

Examiner insight

Examiners reward clear reasoning. When estimating, explicitly state the perfect squares or cubes you are using, for example, 'Since 49 < 55 < 64, we know that √49 < √55 < √64'.

Common pitfall

Forgetting the common perfect cubes (1, 8, 27, 64, 125). Students often know their squares but neglect to learn the cubes, which are essential for estimating cube roots.

Worked example 13 marks

Estimate the value of √70. State which two integers it lies between and which integer it is closer to.

  1. 1

    Find the perfect squares on either side of 70.

  2. 2

    We know that 8² = 64 and 9² = 81.

  3. 3

    So, 64 < 70 < 81.

  4. 4

    Taking the square root of each part gives: √64 < √70 < √81.

  5. 5

    This means 8 < √70 < 9. The two integers are 8 and 9.

  6. 6

    To find which it is closer to, compare the distances: 70 - 64 = 6 and 81 - 70 = 11.

  7. 7

    Since 70 is closer to 64 than to 81, √70 is closer to 8 than to 9.

Worked example 22 marks

Estimate the value of ∛100. State which two integers it lies between.

  1. 1

    Find the perfect cubes on either side of 100.

  2. 2

    We know that 4³ = 64 and 5³ = 125.

  3. 3

    So, 64 < 100 < 125.

  4. 4

    Taking the cube root of each part gives: ∛64 < ∛100 < ∛125.

  5. 5

    This means 4 < ∛100 < 5. The two integers are 4 and 5.

Recap

  • To estimate √x, find the two perfect squares that x lies between.
  • The square root of x will lie between the roots of those perfect squares.
  • To estimate ∛x, find the two perfect cubes that x lies between.
  • To find which integer the root is closer to, see which perfect square/cube the original number is closer to.
  • Memorise the first few square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
  • Memorise the first few cube numbers: 1, 8, 27, 64, 125.

Quick check

  1. Between which two integers does √150 lie?1 mark
  2. Is ∛20 closer to 2 or 3?1 mark

End-of-chapter exercise

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

  1. Calculate the value of 4⁻³. Write your answer as a fraction.2 marks
  2. From the list, identify all the irrational numbers: √100, -5, π, 3/8, √8, 0.666...2 marks
  3. Simplify (5⁴ × 5⁻⁷) / 5⁻¹. Leave your answer as a single power of 5.3 marks
  4. Find the two consecutive integers that √110 lies between.2 marks
  5. The area of a square piece of paper is 70 cm². Estimate the length of one of its sides to the nearest whole number.3 marks
  6. Write (2⁻³)⁻² as a single power of 2. Then, calculate its value.3 marks
  7. Given that x is an integer, find the value of x that satisfies the inequality x < √90 < x + 1.3 marks
  8. A cube has a volume of 150 cm³. Estimate the length of one of its edges, stating which integer it is closest to. You must show your working.4 marks
  9. Show that (√5)² is a rational number but that √5 itself is an irrational number.3 marks
  10. Simplify (81 × 3⁻⁵) ÷ (9⁻²). Express your answer as a single power of 3.4 marks

Go deeper

Practise and revise with member-only material for this chapter.

Workbook

Exam-style practice questions with full worked solutions.

Unlock with a plan

Chapter at a Glance

A single-page summary of every key idea for fast revision.

Unlock with a plan

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Number (9Np)Number (9Nf)Algebra (9Ae)Algebra (9As)Geometry and Measure (9Gg)Geometry and Measure (9Gp)