IGCSE Maths · Number · Core & Extended

Indices and Surds: Laws of Powers and Working with Roots

This note builds directly on Number Basics — you’ll be using your square/cube recall and your sense of rational vs irrational numbers constantly here. Indices let us write repeated multiplication compactly; surds let us keep answers exact instead of rounding. Fractional indices and the whole Surds section are Extended-only — tagged clearly below — but everything else applies to both tiers.

Why this matters A single bacterium in a petri dish doubles every 20 minutes. After just 8 hours, one bacterium becomes 2²⁴ — over 16 million. Nobody writes that out as 2 × 2 × 2… twenty-four times; they write 2²⁴ and evaluate it. Every time something doubles, halves, compounds, or decays repeatedly — population growth, radioactive decay, compound interest, even how fast a rumour spreads — indices are the language used to describe it. Getting comfortable with the laws in this note is what makes all of that tractable later.

1. What Is an Index? Understanding Powers

An index (plural: indices), also called a power or exponent, tells you how many times to multiply a number by itself.

aⁿ = a × a × a × … × a  (n times)

In aⁿ, a is the base and n is the index. Once you can see indices as “repeated multiplication in disguise,” every law of indices becomes something you can work out from scratch rather than something you have to memorise blindly.

Worked example Write 3 × 3 × 3 × 3 × 3 using index notation, then evaluate it.
3 × 3 × 3 × 3 × 3 = 3⁵ = 243
Try it — Warm-up
  1. Write 2 × 2 × 2 × 2 using index notation.
  2. Evaluate 5³.
  3. Evaluate 10⁴.
Show answers
  1. 2⁴
  2. 125
  3. 10000

2. Multiplying and Dividing Powers of the Same Base

Law 1 — Multiplying powers

aᵐ × aⁿ = aᵐ⁺ⁿ

Why? aᵐ is “a multiplied by itself m times,” and aⁿ is “a multiplied by itself n times.” Put them together and you get a multiplied by itself (m + n) times.

Worked example Simplify 3² × 3⁴.
Add the indices: 2 + 4 = 6, so the answer is 3⁶ = 729

Law 2 — Dividing powers

aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Worked example Simplify 5⁶ ÷ 5².
Subtract the indices: 6 − 2 = 4, so the answer is 5⁴ = 625
Try it — Laws 1 & 2
  1. Simplify 4³ × 4².
  2. Simplify 2⁷ ÷ 2³.
  3. Simplify x⁵ × x³ (leave in index form).
  4. Simplify y⁹ ÷ y⁴ (leave in index form).
  5. Simplify 3⁴ × 3 ÷ 3².
Show answers
  1. 4⁵ = 1024
  2. 2⁴ = 16
  3. x⁸
  4. y⁵
  5. 3³ = 27

3. Power of a Power, and the Zero Index

Law 3 — Raising a power to a power

(aᵐ)ⁿ = aᵐⁿ
Worked example Simplify (4²)³.
Multiply the indices: 2 × 3 = 6, so the answer is 4⁶ = 4096

Law 4 — The zero index

a⁰ = 1  (for any a ≠ 0)

This one surprises people until you see it from the division law: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But anything divided by itself is 1 — so a⁰ must equal 1.

Try it — Laws 3 & 4
  1. Simplify (2³)⁴.
  2. Simplify (5²)².
  3. Evaluate 9⁰.
  4. Evaluate 7 × 3⁰.
  5. Simplify (x⁴)³.
Show answers
  1. 2¹² = 4096
  2. 5⁴ = 625
  3. 1
  4. 7 (since 3⁰ = 1)
  5. x¹²

4. Negative Indices

Law 5 — Negative indices

a⁻ⁿ = 1/aⁿ

A negative index doesn’t mean the answer is negative — it means “reciprocal.” This connects straight back to what you already know about reciprocals from Number Basics.

Worked example Find the value of 6⁻².
6⁻² = 1/6² = 1/36
Worked example — combining laws Simplify 3⁻² × 3⁵.
Add the indices: −2 + 5 = 3, so the answer is 3³ = 27
Try it — Negative Indices
  1. Evaluate 4⁻².
  2. Evaluate 2⁻⁵.
  3. Simplify 5⁴ × 5⁻⁶.
  4. Simplify 2⁻³ ÷ 2⁻¹.
  5. Simplify (2⁻¹)³.
Show answers
  1. 1/16
  2. 1/32
  3. 5⁻² = 1/25
  4. 2⁻² = 1/4
  5. 2⁻³ = 1/8

5. Fractional Indices Extended only

A fractional index is just another way of writing a root.

a^(1/n) = ⁿ√a
a^(m/n) = (ⁿ√a)ᵐ
Worked examples • 49^(1/2) = √49 = 7
• 8^(1/3) = ³√8 = 2
• 27^(2/3) = (³√27)² = 3² = 9
• 16^(−1/2) = 1/(16^(1/2)) = 1/√16 = 1/4
Try it — Fractional Indices Extended
  1. Evaluate 25^(1/2).
  2. Evaluate 64^(1/3).
  3. Evaluate 8^(2/3).
  4. Evaluate 100^(−1/2).
  5. Evaluate 32^(3/5).
Show answers
  1. 5
  2. 4
  3. 4 (³√8 = 2, then 2² = 4)
  4. 1/10
  5. 8 (⁵√32 = 2, then 2³ = 8)

6. Combining All the Laws

Exam questions rarely use one law in isolation — the real skill is spotting which law to apply first.

Worked example Simplify (2x³)⁴.
Apply the power to everything inside the bracket: 2⁴ × x³ˣ⁴ = 16 × x¹² = 16x¹²
Worked example Simplify 12a⁷ ÷ 4a⁻³.
Divide the numbers: 12 ÷ 4 = 3. Subtract the indices: 7 − (−3) = 10.
Answer: 3a¹⁰
Try it — Mixed Index Laws
  1. Simplify (3x²)³.
  2. Simplify 20a⁶ ÷ 5a⁻²​.
  3. Solve for x: 2ˣ = 128.
  4. Solve for x: 5ˣ⁺¹ = 25.
  5. Simplify 6a⁴b² × 2a⁻¹b⁵.
Show answers
  1. 27x⁶
  2. 4a⁸
  3. x = 7 (since 2⁷ = 128)
  4. x = 1 (since 5² = 25, so x + 1 = 2)
  5. 12a³b⁷

7. Solving Simple Index Equations

When the unknown is in the power, the strategy is almost always the same: write both sides as powers of the same base, then set the indices equal to each other.

Worked example — bases already match Solve 3ˣ = 243.
Write 243 as a power of 3: 243 = 3⁵.
So 3ˣ = 3⁵, which means x = 5.
Worked example — converting to a common base first Solve 9ˣ = 27.
Both 9 and 27 are powers of 3: 9 = 3², 27 = 3³.
So (3²)ˣ = 3³ → 3²ˣ = 3³ → 2x = 3 → x = 3/2
Worked example — unknown appears twice Solve 2^(2x−1) = 32.
32 = 2⁵, so 2x − 1 = 5 → 2x = 6 → x = 3
Try it — Index Equations
  1. Solve 2ˣ = 32.
  2. Solve 5ˣ = 125.
  3. Solve 4ˣ = 64.
  4. Solve 3^(x+1) = 81.
  5. Solve 2^(3x) = 16. Ext.
Show answers
  1. x = 5
  2. x = 3
  3. x = 3 (since 4³ = 64)
  4. x = 3 (since 3⁴ = 81, so x + 1 = 4)
  5. x = 4/3 (since 16 = 2⁴, so 3x = 4)

8. What Is a Surd? Extended only

A surd is a root that can’t be simplified to a whole number — it’s an irrational number left in root form on purpose, because writing it as a decimal would mean rounding and losing accuracy. √2, √5, and √11 are all surds. √16 is not a surd, because it simplifies to the whole number 4.

This is exactly the rational/irrational distinction from Number Basics, applied directly: a surd is simply an irrational square root written in exact form.

9. Simplifying Surds Extended only

√(a × b) = √a × √b

To simplify a surd, look for the largest square number hiding inside it as a factor.

Worked example Simplify √98.
98 = 49 × 2, and 49 is a square number.
√98 = √49 × √2 = 7√2
Worked example — combining like surds Simplify √45 + √80.
√45 = √9 × √5 = 3√5
√80 = √16 × √5 = 4√5
3√5 + 4√5 = 7√5

Note: you can only add or subtract surds that simplify to the same root — 3√5 + 4√5 works, but 3√5 + 4√2 cannot be combined any further.

Try it — Simplifying Surds Extended
  1. Simplify √showing working: √32.
  2. Simplify √180.
  3. Simplify √63 − √28.
  4. Simplify √12 + √27.
  5. Simplify √8 × √2.
Show answers
  1. 4√2
  2. 6√5
  3. √63 = 3√7, √28 = 2√7 → 3√7 − 2√7 = √7
  4. √12 = 2√3, √27 = 3√3 → 2√3 + 3√3 = 5√3
  5. √8 × √2 = √16 = 4

10. Rationalising the Denominator Extended only

Mathematicians don’t leave a surd on the bottom of a fraction — it’s considered “untidy” and makes fractions harder to compare. To remove it, multiply top and bottom by something that turns the denominator into a whole number.

Worked example — simple denominator Rationalise 12/√6.
Multiply top and bottom by √6: (12√6)/(√6 × √6) = (12√6)/6 = 2√6
Worked example — binomial denominator (using the conjugate) Rationalise 1/(3 − √2).
Multiply top and bottom by the conjugate (3 + √2):
(3 + √2) / [(3−√2)(3+√2)] = (3 + √2) / (9 − 2) = (3 + √2)/7
Try it — Rationalising Extended
  1. Rationalise 15/√5.
  2. Rationalise 4/√3.
  3. Rationalise 2/(5 + √3).
  4. Rationalise 6/(√7 − 2).
  5. Expand and simplify (4 + √3)(4 − √3). What do you notice about the surd?
Show answers
  1. 3√5
  2. (4√3)/3
  3. 2(5 − √3)/22 = (5 − √3)/11
  4. 6(√7 + 2)/3 = 2(√7 + 2) = 2√7 + 4
  5. 16 − 3 = 13 — the surd cancels out completely, because (a+√b)(a−√b) = a² − b. This is exactly why conjugates work for rationalising.

11. Connecting Surds and Fractional Indices Extended only

Section 5 showed a^(1/n) = ⁿ√a — surds and fractional indices are two notations for exactly the same thing. Being able to switch between them fluently is genuinely useful: some expressions are much easier to simplify in index form, others in root form.

Root (surd) formIndex form
√aa^(1/2)
³√aa^(1/3)
(³√a)²a^(2/3)
1/√aa^(−1/2)
Worked example Write √7 × ⁴√7 as a single power of 7, then evaluate 7¹ if possible.
√7 = 7^(1/2), ⁴√7 = 7^(1/4). Multiplying powers of the same base: add the indices.
1/2 + 1/4 = 3/4, so √7 × ⁴√7 = 7^(3/4) (this doesn’t simplify to a whole number, which makes sense — it’s still a surd in disguise).
Try it — Surd/Index Conversion Extended
  1. Write √11 in index form.
  2. Write 5^(1/3) in root form.
  3. Write (⁵√2)³ in index form.
  4. Simplify √3 × √3 using index laws: write both as 3^(1/2), add the indices, then evaluate.
Show answers
  1. 11^(1/2)
  2. ³√5
  3. 2^(3/5)
  4. 3^(1/2) × 3^(1/2) = 3^(1/2+1/2) = 3¹ = 3

12. Estimating the Value of a Surd Extended only

Sometimes you just need a sensible estimate of a surd, without a calculator — useful for sanity-checking an answer. The trick is to “sandwich” the number between two square numbers you already know.

Worked example Between which two consecutive whole numbers does √60 lie?
Find the square numbers either side of 60: 49 (= 7²) and 64 (= 8²).
Since 49 < 60 < 64, it follows that 7 < √60 < 8. So √60 lies between 7 and 8 — closer to 8, since 60 is nearer to 64 than to 49.
Try it — Estimating Surds Extended
  1. Between which two whole numbers does √40 lie?
  2. Between which two whole numbers does √90 lie?
  3. Between which two whole numbers does √130 lie?
  4. Without a calculator, decide whether √70 is closer to 8 or 9.
Show answers
  1. Between 6 and 7 (36 < 40 < 49)
  2. Between 9 and 10 (81 < 90 < 100)
  3. Between 11 and 12 (121 < 130 < 144)
  4. Closer to 8 — 70 is much nearer to 64 (8²) than to 81 (9²).
Common mistakes to watch for
  • Writing aᵐ × aⁿ as a^(mn) instead of a^(m+n) — multiplying powers means adding indices, not multiplying them.
  • Treating a negative index as a negative answer — a⁻ⁿ is a positive reciprocal, e.g. 2⁻³ = 1/8, not −8.
  • Forgetting the power applies to a coefficient too — (3x²)² = 9x⁴, not 3x⁴.
  • Adding surds by adding the numbers under the root — √a + √b is not √(a+b). Simplify first, then only combine surds with matching roots.
  • Stopping halfway through rationalising a binomial denominator — you must multiply by the full conjugate, and expand both top and bottom carefully.

13. Quick-Fire Challenge Round

Fast instinct-checking round — no long working, just answer and verify.

True or False?
  1. a⁰ = 0 for any value of a.
  2. A negative index always gives a negative answer.
  3. √50 is a surd.
  4. √a × √a = a for any positive number a.
  5. You can add √3 and √12 once both are simplified, because they share the same root.
  6. Rationalising a denominator changes the value of the fraction.
Show answers
  1. False — a⁰ = 1, not 0.
  2. False — a negative index means “reciprocal,” not “negative value.” E.g. 2⁻² = 1/4, which is positive.
  3. True — 50 isn’t a perfect square, so √50 doesn’t simplify to a whole number.
  4. True — that’s the definition of a square root.
  5. True — √12 simplifies to 2√3, so √3 + √12 = √3 + 2√3 = 3√3. Ext.
  6. False — multiplying top and bottom by the same value keeps the fraction’s value exactly the same; it only changes how it looks. Ext.
Spot the Error
  1. “3² × 3⁴ = 3⁸” (add or multiply the indices here?)
  2. “5⁻² = −25”
  3. “√18 + √2 = √20”
  4. “(2x²)³ = 2x⁶”
Show answers
  1. Indices are added when multiplying powers of the same base: 2 + 4 = 6, so the answer is 3⁶, not 3⁸.
  2. 5⁻² = 1/5² = 1/25, not −25. A negative index means reciprocal, not a negative result.
  3. You can’t add surds by adding what’s inside the root — simplify first: √18 = 3√2, so √18 + √2 = 3√2 + √2 = 4√2, not √20 (although coincidentally √20 = 2√5, which isn’t even the same value).
  4. The power applies to the number too: (2x²)³ = 2³ × x⁶ = 8x⁶, not 2x⁶.

14. Mixed Practice — Bringing It All Together

Core & Extended
  1. Simplify 6⁵ ÷ 6².
  2. Simplify (4²)³.
  3. Evaluate 5⁻³.
  4. Simplify 9x⁴ × 2x⁻¹.
  5. Solve for x: 3ˣ = 81.
  6. Solve for x: 5ˣ = 625.
Show answers
  1. 6³ = 216
  2. 4⁶ = 4096
  3. 1/125
  4. 18x³
  5. x = 4
  6. x = 4
Extended-level mixed set Extended only
  1. Evaluate 36^(1/2) + 8^(1/3).
  2. Simplify √200 − √72.
  3. Rationalise 10/√2.
  4. Evaluate 27^(−2/3).
  5. Simplify (2√3)².
  6. Between which two whole numbers does √85 lie?
Show answers
  1. 6 + 2 = 8
  2. √200 = 10√2, √72 = 6√2 → 10√2 − 6√2 = 4√2
  3. 5√2
  4. (³√27)⁻² = 3⁻² = 1/9
  5. (2√3)² = 4 × 3 = 12
  6. Between 9 and 10 (81 < 85 < 100)

Practice at Home

A large set again — work through it steadily over a few sittings rather than in one go.

Part A — Understanding indices & multiplying/dividing powers
  1. Write 7 × 7 × 7 using index notation and evaluate it.
  2. Simplify 6⁴ × 6³.
  3. Simplify 9⁸ ÷ 9⁵.
  4. Simplify p⁶ × p² ÷ p³ (leave in index form).
Part B — Power of a power, zero and negative indices
  1. Simplify (3⁴)².
  2. Evaluate 15⁰.
  3. Evaluate 3⁻⁴.
  4. Simplify 7² × 7⁻⁵.
  5. Simplify (5⁻²)² without a calculator.
Part C — Fractional indices Extended
  1. Evaluate 121^(1/2).
  2. Evaluate 1000^(1/3).
  3. Evaluate 4^(5/2).
  4. Evaluate 81^(−1/4).
Part D — Solving index equations
  1. Solve 2ˣ = 64.
  2. Solve 7ˣ = 343.
  3. Solve 8ˣ = 2 (hint: write 8 as 2³, and 2 as 2¹). Ext.
  4. Solve 3^(2x+1) = 27. Ext.
Part E — Mixed index laws
  1. Simplify (5x³)².
  2. Simplify 24a⁵b³ ÷ 6a²b⁻¹.
  3. Solve for x: 4ˣ = 256.
  4. Solve for x: 2ˣ⁻¹ = 32.
Part F — Simplifying and combining surds Extended
  1. Simplify √48.
  2. Simplify √125.
  3. Simplify √54 + √24.
  4. Simplify √99 − √44.
  5. Simplify √5 × √20.
Part G — Rationalising denominators Extended
  1. Rationalise 9/√3.
  2. Rationalise 14/√7.
  3. Rationalise 5/(2 + √3).
  4. Rationalise 3/(√5 − 1).
Part H — Estimating surds and surd/index conversion Extended
  1. Between which two whole numbers does √55 lie?
  2. Between which two whole numbers does √115 lie?
  3. Write ³√10 in index form.
  4. Write 6^(2/3) in root form.
Part I — Challenge Extended
  1. Show that (1 + √3)² = 4 + 2√3.
  2. Solve 2^(x) × 2^(x+1) = 32 for x.
  3. Simplify (√8 + √2)², giving your answer in the form a + b√c.

Key Vocabulary Recap

Index / power / exponent — tells you how many times to multiply a number by itself
Base — the number being raised to a power
Reciprocal — what a negative index gives you: 1 over the positive power
Common base — writing both sides of an equation as powers of the same number, to compare indices directly
Surd — a root that can’t be simplified to a whole number
Rationalise — remove a surd from the denominator of a fraction Ext.
Conjugate — flip the sign between two terms, e.g. (a + √b) → (a − √b) Ext.

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C1.7 (Core subject content — Indices I) and sections E1.7 and E1.18 (Extended subject content — Indices I, Surds). Core indices work covers positive, zero, and negative integer powers only; Extended adds fractional indices and the full Surds topic, both marked “Extended only” above.

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