Powers and roots: reversing a process
Connect powers to repeated multiplication and roots to the question that reverses it.
Core and Extended. Fractional and negative index laws belong to section 1.7 and will follow later.
Before you begin
- Multiply whole numbers.
- Know that multiplying two negative numbers gives a positive result.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does 3⁴ mean?
What you will learn
- Calculate squares, cubes and other whole-number powers.
- Find roots and distinguish a principal root from equation solutions.
- Use inverse operations to solve area and volume problems.
A power counts factors
In 4³, the base is 4 and the exponent is 3. The expression means 4 × 4 × 4 = 64. It does not mean 4 × 3. Squaring uses two equal factors; cubing uses three. A square of side 4 has area 16 square units; a cube of side 4 has volume 64 cubic units. The exponent matches the number of length dimensions being multiplied.
Other powers work in the same way: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Before reaching for a calculator, identify the repeated factor and how many copies are needed. If the base is negative, brackets tell you whether the sign is part of each factor.
A root asks for the original factor
The square root of 36 asks for the non-negative number whose square is 36, so √36 = 6. The cube root of 64 asks for the number whose cube is 64, so ∛64 = 4. The fourth root of 81 is 3 because 3⁴ = 81. Check any root by raising the result to the matching power.
Odd roots of negative numbers are real: ∛(−27) = −3 because (−3)³ = −27. Even powers of real numbers are non-negative, so a negative number has no real square root. Within this course, √(−9) is not a real number.
One root symbol; sometimes two equation solutions
The radical symbol √ gives the principal, non-negative square root. Thus √25 = 5, not ±5. However, the equation x² = 25 has two real solutions, x = 5 and x = −5, because either value squares to 25. Read whether you are evaluating an expression or solving an equation.
Brackets also change the order: (−5)² = (−5) × (−5) = 25, while −5² means −(5²) = −25. The minus sign in the second expression is applied after the power. Writing the repeated multiplication is a reliable way to see the difference.
Build recall with checks
Practise the squares of 1 through 15 and their matching roots: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Read this list both ways: 12² = 144 and √144 = 12. The gap between consecutive squares grows by 2: 11² = 10² + 21 = 121.
The cubes to recall here are 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125 and 10³ = 1000, together with the reverse cube-root facts. Recall supports reasoning: since 6² = 36 and 7² = 49, √40 lies between 6 and 7. This checks scale even when you do not know the exact decimal.
Pause and explain
A student writes √64 = ±8. What should change?
Worked example
A cube has volume 343 cm³. Find its side length, then its total surface area.
Show the worked solution
- Volume = side³, so find ∛343. Since 7 × 7 × 7 = 343, the side is 7 cm.
- Each of the six square faces has area 7² = 49 cm².
- Total surface area = 6 × 49 = 294 cm². Check that the volume uses cm³ and surface area uses cm².
Answer Side 7 cm; surface area 294 cm².
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.
Calculate ∛125 + √144.
Give me a hint
Find the number whose cube is 125 and the non-negative number whose square is 144.
Compare my reasoning
- 5³ = 125, so ∛125 = 5.
- 12² = 144, so √144 = 12.
- Add after taking the roots: 5 + 12 = 17.
17
Look for these in your work
- Used the correct inverse for each power.
- Did not take a single root of the sum.
Find 2⁵, the fourth root of 81, and all real solutions of x² = 121.
Give me a hint
Write out repeated multiplication where needed; the last item is an equation.
Compare my reasoning
- 2⁵ = 32.
- 3⁴ = 81, so the fourth root of 81 is 3.
- Both 11² and (−11)² are 121, so x = 11 or −11.
32; 3; x = ±11.
Look for these in your work
- Separated evaluating a root from solving an equation.
- Checked by reversing each operation.
A square photograph has area 196 cm². A border 2 cm wide is added outside every edge. Find the area of the border alone.
Give me a hint
Recover the photo's side. The border adds 2 cm at both ends of each dimension.
Compare my reasoning
- The photo side is √196 = 14 cm.
- The outside side is 14 + 2 + 2 = 18 cm.
- Border area = 18² − 14² = 324 − 196 = 128 cm².
128 cm²
Look for these in your work
- Used the square root to recover a length.
- Added the border on both sides.
- Subtracted the original area to isolate the border.
Common mistakes
- Multiplying the base by the exponent instead of multiplying repeated factors.
- Writing ±6 as the value of √36.
- Using a square root to undo a cube.
What can you explain now?
Evaluate (−4)² and −4². Explain the difference.
Compare with the explanation
16 and −16.
The first squares the entire negative number. In the second, the power is evaluated before applying the leading minus sign.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Without notes, explain why √100 = 10 but x² = 100 has two solutions. Then recall 13², 14² and 15².
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources