Order values and use operations and index laws
Compare signed numbers, follow operation order and simplify integer and fractional powers with the correct index laws.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 5⁰ for non-zero 5?
What you will learn
- Order quantities and interpret equality and inequality symbols
- Calculate with signed integers, fractions and decimals using brackets
- Apply index laws, including fractional indices for Extended
Order operations and preserve signs
Calculate brackets first, then powers, multiplication and division from left to right, then addition and subtraction from left to right. A negative number in a power needs brackets if the sign is part of the base.
On a number line, numbers farther right are greater. For example, −2 > −5. The symbols ≥ and ≤ include equality; > and < do not.
Index laws keep a common base visible
For the same non-zero base, aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ. Also a⁰ = 1 and a⁻ⁿ = 1/aⁿ. Core uses positive, zero and negative integer indices; Extended also uses fractional indices such as a^(1/2) = √a.
Do not add exponents when the bases differ. Reduce a fractional index to its root and power meaning before evaluating it.
Pause and explain
What is 2⁻³?
Worked example
Evaluate −3 + 2(5 − 8)² and simplify 2³ × 2⁻⁵ ÷ 2⁻¹.
Show the worked solution
- Inside the brackets, 5 − 8 = −3; its square is 9.
- Then −3 + 2 × 9 = 15.
- For the common base 2, add then subtract exponents: 3 − 5 − (−1) = −1, so the value is 1/2.
Answer −3 + 2(5 − 8)² = 15; 2³ × 2⁻⁵ ÷ 2⁻¹ = 1/2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Simplify (3² × 3⁻¹) ÷ 3⁰, then evaluate (−4)² and −4².
Give me a hint
Use exponent rules only on powers with the same base, and track brackets around the negative base.
Compare my reasoning
- (3² × 3⁻¹) ÷ 3⁰ = 3^(2−1−0) = 3.
- (−4)² = 16 because the negative sign is part of the base.
- −4² = −16 because the power is evaluated before the leading negative sign.
3; (−4)² = 16; −4² = −16.
Look for these in your work
- Index laws applied to like bases
- Bracketed negative squared
- Unbracketed negative interpreted correctly
Evaluate 3/4 − 2(−1/2)² and simplify 16^(3/4).
Give me a hint
Square the fraction before subtracting; write the fractional index as a root followed by a power.
Compare my reasoning
- 2(−1/2)² = 2 × 1/4 = 1/2.
- 3/4 − 1/2 = 1/4.
- 16^(3/4) = (⁴√16)³ = 2³ = 8.
3/4 − 2(−1/2)² = 1/4; 16^(3/4) = 8.
Look for these in your work
- Order of operations retained
- Fraction subtraction correct
- Extended fractional index interpreted
A diver is at −8 m and rises 3 m, then descends 5 m. Write the final position and order it against −6 m.
Give me a hint
Treat upward and downward movement as signed changes from the starting position.
Compare my reasoning
- Final position is −8 + 3 − 5 = −10 m.
- Compare −10 and −6 on a number line.
- −10 is farther left, so −10 < −6.
The diver is at −10 m, which is less than −6 m.
Look for these in your work
- Movement signs match the directions
- Starting position included
- Negative numbers ordered correctly
Common mistakes
- Changing the sign when ordering negative numbers
- Adding indices across unlike bases
- Applying an index law to a sum
- Treating −a² as (−a)²
What can you explain now?
Which is greater, −2 or −5?
Compare with the explanation
−2
On a number line, −2 is to the right of −5.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What do negative and fractional indices mean, and when must brackets include a negative sign?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.