Use powers, roots and equivalent fraction forms
Recall common squares and cubes, simplify fractions, and convert between fractions, decimals and percentages.
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
Which value is equivalent to 3/5?
What you will learn
- Calculate common squares, square roots, cubes and cube roots
- Convert mixed numbers, improper fractions, decimals and percentages
- Reduce fractions to their simplest form
Recognise powers and roots as inverse operations
A square root reverses squaring: √169 = 13 because 13² = 169. A cube root reverses cubing: ∛1000 = 10 because 10³ = 1000. Know squares and roots for integers 1–15, and cubes and roots for 1, 2, 3, 4, 5 and 10.
A power may be larger than a square or cube. Use brackets carefully: (−4)² = 16 but −4² = −16 under the usual order of operations.
Equivalent fractions keep the same value
To simplify a fraction, divide its numerator and denominator by their highest common factor. To convert a fraction to a decimal, divide the numerator by the denominator; multiply the decimal by 100 to write it as a percentage.
For a mixed number, convert to an improper fraction before multiplying or dividing. A percentage greater than 100% is valid and represents more than the whole starting quantity.
Pause and explain
What is √225?
Worked example
Write 3/8 as a decimal and percentage, and evaluate √169 and ∛1000.
Show the worked solution
- 3 ÷ 8 = 0.375.
- 0.375 × 100 = 37.5%.
- 13² = 169 and 10³ = 1000, so the exact roots are 13 and 10.
Answer 3/8 = 0.375 = 37.5%; √169 = 13; ∛1000 = 10.
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.
Convert 7/20 to a decimal and percentage, then simplify 18/24.
Give me a hint
Divide for the decimal; cancel the HCF for the simplified fraction.
Compare my reasoning
- 7 ÷ 20 = 0.35.
- 0.35 × 100 = 35%.
- The HCF of 18 and 24 is 6, so 18/24 = 3/4.
7/20 = 0.35 = 35%; 18/24 = 3/4.
Look for these in your work
- Decimal division correct
- Percentage conversion correct
- Fraction reduced fully
Write 2 3/5 as an improper fraction, decimal and percentage.
Give me a hint
Convert the mixed number to fifths, then convert the value 2.6.
Compare my reasoning
- 2 3/5 = (2 × 5 + 3)/5 = 13/5.
- 13 ÷ 5 = 2.6.
- 2.6 × 100 = 260%.
2 3/5 = 13/5 = 2.6 = 260%.
Look for these in your work
- Mixed number converted correctly
- Decimal agrees with the fraction
- Percentage may exceed 100%
A square tile has area 196 cm². Find its side length, then express that length as a fraction of a 2 m board and as a percentage.
Give me a hint
Find the positive square root; convert both lengths to the same unit before forming a fraction.
Compare my reasoning
- The tile side is √196 = 14 cm.
- 2 m = 200 cm, so the fraction is 14/200 = 7/100.
- 7/100 = 7%.
Side length 14 cm; it is 7/100, or 7%, of the board length.
Look for these in your work
- Positive length selected
- Units matched
- Fraction and percentage agree
Common mistakes
- Multiplying by 100 when converting a percentage back to a decimal
- Leaving a fraction unsimplified
- Confusing the square root with the two solutions of x² = 169
- Dropping a negative sign outside a power
What can you explain now?
What is 7/20 as a percentage?
Compare with the explanation
35%
7 ÷ 20 = 0.35, then multiply by 100.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you move between a simplified fraction, decimal and percentage without changing the value?
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.