SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 21
Indices compress repeated multiplication. Standard form compresses scale. Together they let Mathematics describe very large and very small quantities without losing structure.
This guide develops powers, index notation, multiplication and division laws, powers of powers, zero indices, negative indices, roots as inverse operations, standard form, scientific notation, estimation and calculator checks. Exact depth and sequencing vary by subject level and school, so extension material should be used only when it matches the learner’s current course.
Useful prior guides: Prime Factorisation, HCF, LCM, Squares, Cubes and Roots, Calculator Skills, Exact Values and Input Discipline and Numbers, Number Lines, Approximation and Estimation.
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1. A power has a base and an index
In 5³, 5 is the base and 3 is the index. The expression means 5×5×5 = 125.
An index tells how many equal factors of the base are multiplied together.
Do not confuse multiplication with exponentiation
5³ is not 5×3. It is 5 multiplied by itself three times.
2. Index notation makes prime structure visible
72 = 2³×3². The exponents record how many copies of each prime factor appear.
This is why indices connect naturally to prime factorisation, HCF, LCM, squares and cubes.
Square test
A positive integer is a perfect square exactly when every prime exponent is even.
3. Multiply same bases by adding indices
a³×a⁵ = a⁸ because the product contains 3+5 copies of a.
Worked example
2⁴×2³ = 2⁷ = 128.
Boundary condition
The bases must match before indices can be combined in this way.
4. Different bases do not combine by adding indices
2³×3² cannot be simplified to 6⁵. The expression means 8×9 = 72, while 6⁵ is 7776.
Index laws preserve repeated-factor structure; they are not shortcuts detached from meaning.
5. Divide same bases by subtracting indices
For non-zero a, a⁷÷a³ = a⁴.
Reason: a⁷/a³ contains seven factors of a above and three below. Cancelling three leaves four.
Worked example
5⁶÷5² = 5⁴ = 625.
6. A power raised to a power multiplies indices
(a³)⁴ = a¹² because four copies of a³ contain 3×4 copies of a.
Worked example
(2³)⁴ = 2¹² = 4096.
Do not add the indices here. The structure is repeated groups of repeated factors.
7. A product inside brackets distributes the outside power
(ab)³ = a³b³.
Worked example
(2x)³ = 8x³.
The coefficient is also cubed: 2³ = 8.
8. A quotient inside brackets distributes the outside power
(a/b)³ = a³/b³, provided b ≠ 0.
Worked example
(2/5)³ = 8/125.
This follows because (2/5)(2/5)(2/5) multiplies numerators and denominators separately.
9. Zero index comes from the division law
For non-zero a:
a⁵÷a⁵ = 1.
Using the index law, a⁵÷a⁵ = a⁰. Therefore a⁰ = 1 for a ≠ 0.
Worked example
7⁰ = 1.
10. Negative indices represent reciprocals
For non-zero a, a⁻³ = 1/a³.
Why?
a²÷a⁵ = a⁻³ by subtracting indices. Cancelling factors directly gives 1/a³.
Worked example
2⁻⁴ = 1/16.
If negative indices have not yet appeared in the learner’s current course, treat this section as extension.
11. Negative bases need brackets
(−3)² = 9, while −3² conventionally means −(3²) = −9.
When the negative sign belongs to the base, include it inside brackets.
Odd powers preserve the negative sign
(−2)³ = −8.
12. Roots reverse powers
√81 = 9 because 9² = 81. ∛125 = 5 because 5³ = 125.
Indices and roots are inverse structures. Prime factorisation can make the relationship visible.
Example
√(2⁶×3²) = 2³×3 = 24.
13. Fractional indices connect powers and roots
An extension relationship is a^(1/2) = √a for suitable real a, and a^(1/3) = ∛a.
For example, 25^(1/2) = 5.
Treat fractional indices as extension unless they are explicitly part of the learner’s current course.
14. Standard form writes scale as a coefficient times a power of ten
Standard form is written as:
A × 10ⁿ, where 1 ≤ |A| < 10 for non-zero values.
Examples
56,000 = 5.6×10⁴.
0.00072 = 7.2×10⁻⁴.
15. Positive powers of ten move scale upward
4.3×10⁵ = 430,000.
The exponent 5 means multiply by 100,000.
Do not rely only on “move the decimal”
The safer meaning is multiplication by a power of ten. Decimal movement is the visual consequence.
16. Negative powers of ten describe small magnitudes
6.2×10⁻³ = 0.0062.
Because 10⁻³ = 1/1000.
Magnitude check
A negative power of ten should produce a value smaller than the coefficient in magnitude when the coefficient is finite and non-zero.
17. Multiply numbers in standard form by separating coefficient and power
(3×10⁴)(2×10³) = 6×10⁷.
Renormalise if necessary
(6×10⁵)(4×10²) = 24×10⁷ = 2.4×10⁸.
The final coefficient must be at least 1 and less than 10 in magnitude.
18. Divide numbers in standard form by subtracting powers
(8×10⁷)/(2×10³) = 4×10⁴.
Worked example
(9×10⁵)/(3×10⁻²) = 3×10⁷.
Because 10⁵÷10⁻² = 10^(5−(−2)) = 10⁷.
19. Add and subtract standard-form numbers only after matching scale
3.2×10⁵ + 4.7×10⁴.
Rewrite 4.7×10⁴ as 0.47×10⁵.
Then total = 3.67×10⁵.
You cannot simply add 3.2 and 4.7 while ignoring different powers of ten.
20. Estimation remains essential
(4.98×10⁶)(2.02×10³) is close to (5×10⁶)(2×10³) = 10×10⁹ = 10¹⁰.
The exact product should therefore be near 1×10¹⁰.
An answer such as 1×10⁷ would fail the magnitude check.
21. Calculator displays may use E notation
Some calculators display 6.3E5 to mean 6.3×10⁵.
Read the display as scientific notation, not as an algebraic variable.
Check the sign of the exponent carefully; 6.3E−5 is very different from 6.3E5.
22. Common index errors
| Error | Why it fails | Repair prompt |
|---|---|---|
| a²+a³ = a⁵ | Index law applies to multiplication, not addition | Are the terms being multiplied? |
| (a²)³ = a⁵ | Powers of powers multiply indices | How many copies of a appear in total? |
| a⁰ = 0 | Zero index comes from equal-factor division | What is a⁵/a⁵? |
| 4.8×10³ + 2.2×10² = 7×10⁵ | Scales were not matched | Can both terms be written with the same power of ten? |
| 14×10⁶ left as standard form | Coefficient outside required range | Can the coefficient be renormalised? |
23. Practice laboratory
- Write 4×4×4×4 using index notation.
- Evaluate 3⁴.
- Simplify x³×x⁵.
- Simplify a⁹÷a⁴.
- Simplify (m³)⁴.
- Simplify (2x)³.
- Evaluate 7⁰.
- Write 2⁻³ as a fraction.
- Write 720,000 in standard form.
- Write 0.000093 in standard form.
- Write 5.4×10⁶ as an ordinary number.
- Write 8.2×10⁻⁴ as an ordinary decimal.
- Evaluate (3×10⁴)(5×10³) in standard form.
- Evaluate (8×10⁹)/(2×10⁴) in standard form.
- Evaluate 3.4×10⁵ + 7.0×10⁴.
- State whether 23×10⁶ is in standard form.
- Estimate the order of magnitude of (6.1×10⁷)(1.9×10⁻³).
24. Explained answers
1. 4⁴.
2. 81.
3. x⁸.
4. a⁵.
5. m¹².
6. 8x³.
7. 1.
8. 1/8.
9. 7.2×10⁵.
10. 9.3×10⁻⁵.
11. 5,400,000.
12. 0.00082.
13. 15×10⁷ = 1.5×10⁸.
14. 4×10⁵.
15. 3.4×10⁵ + 0.7×10⁵ = 4.1×10⁵.
16. No. Rewrite as 2.3×10⁷.
17. About 6×2×10⁴ = 12×10⁴ ≈ 10⁵.
25. Complete mixed problem
A fictional sensor records 4.8×10⁶ units per hour for 3.5×10² hours. Estimate first, then find the total in standard form.
Estimate: 5×10⁶ × 4×10² = 20×10⁸ = 2×10⁹.
Exact model calculation:
(4.8×3.5)×10⁸ = 16.8×10⁸ = 1.68×10⁹.
The exact result is consistent with the estimate.
26. Teaching indices through structure
Before giving laws, expand small examples into repeated factors. Let the learner see why exponents add in products and subtract in quotients.
Then compress back into index notation. The law becomes a summary of structure rather than an isolated rule.
Changed-case test
Ask why x³+x⁴ does not become x⁷ even though x³×x⁴ does. This reveals whether the learner knows which operation activates the law.
27. Questions students often ask
Why is anything to power zero equal to one?
For a non-zero base, it follows from dividing equal powers: aⁿ/aⁿ = a⁰ = 1.
Why does a negative index not make the number negative?
The negative sign belongs to the exponent and indicates a reciprocal, not the sign of the value.
Why standard form?
It makes scale explicit and allows large or small quantities to be compared and calculated efficiently.
Can I use my calculator’s E display?
Yes, if you understand that E represents “×10 to the power”. Follow the notation required in written answers.
28. Return path and sources
Indices connect number structure to algebra and scientific scale. Revisit Prime Factorisation, HCF, LCM, Squares, Cubes and Roots for factor structure and Calculator Skills for reliable entry and estimation.
Official curriculum reference: MOE Secondary Syllabus Directory. Exact treatment of zero, negative and fractional indices and standard form varies by subject level and school.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Expand the repeated structure, apply only the law justified by that structure, preserve scale, estimate magnitude and verify the final form.
