Surds & Index Laws
Master the rules of powers and roots. Learn to classify numbers, apply the six index laws, simplify surds, rationalise denominators, and work with standard form.
What You'll Learn
- Classify numbers as rational or irrational and explain the difference
- Apply all six index laws confidently, including negative and fractional indices
- Simplify surds by extracting perfect square factors
- Add, subtract, and multiply surd expressions
- Rationalise denominators (simple and conjugate methods)
- Convert between standard form and ordinary numbers
IB Assessment Focus
Criterion A: Apply index laws in unfamiliar, multi-step problems.
Criterion B: Discover patterns and justify why each law works.
Criterion C: Communicate your working with correct mathematical language.
Criterion D: Apply surds/indices to real-world contexts (e.g., scientific measurements).
Number Classification
Before working with surds and indices, you must understand how numbers are classified. Every number you encounter belongs to one or more of these categories:
| Type | Definition | Examples |
|---|---|---|
| Natural (ℕ) | Counting numbers from 1 onwards | 1, 2, 3, 47, 1000 |
| Integer (ℤ) | Whole numbers, including zero and negatives | -5, 0, 3, 12 |
| Rational (ℚ) | Any number expressible as p/q where p, q are integers and q ≠ 0 | 3/4, -2, 0.75, 0.333... |
| Irrational | Cannot be written as a fraction; decimal is non-terminating and non-repeating | √2, π, √5, e |
| Real (ℝ) | All rational and irrational numbers combined | All of the above |
The Six Index Laws
Index laws (also called exponent rules) let you simplify expressions involving powers. Every law requires the same base unless you are raising a power to a power.
Law 1 — Multiplication (same base)
When multiplying powers with the same base, add the exponents. This works because a³ × a² means (a × a × a) × (a × a) = a&sup5;.
- 3² × 3&sup4; = 32+4 = 3&sup6; = 729
- x&sup5; × x³ = x5+3 = x&sup8;
- 2 × 2&sup4; = 2¹ × 2&sup4; = 2&sup5; = 32 (remember: 2 = 2¹)
Law 2 — Division (same base)
When dividing powers with the same base, subtract the exponents. This is because dividing cancels out repeated multiplication.
- 5&sup6; ÷ 5² = 56−2 = 5&sup4; = 625
- x&sup8; ÷ x³ = x8−3 = x&sup5;
- 7³ ÷ 7³ = 73−3 = 7° = 1 (this leads to Law 4)
Law 3 — Power of a Power
When raising a power to another power, multiply the exponents. Think of it as repeated exponentiation: (a²)³ = a² × a² × a² = a2+2+2 = a&sup6;.
- (2³)² = 23×2 = 2&sup6; = 64
- (x&sup4;)&sup5; = x4×5 = x20
- (10²)³ = 10&sup6; = 1,000,000
Law 4 — Zero Index
Any non-zero number raised to the power of zero equals 1. Justification: a³ ÷ a³ = a3−3 = a°. But a³ ÷ a³ = 1 (anything divided by itself is 1). Therefore a° = 1.
- 5° = 1
- (-3)° = 1
- (1000)° = 1
- x° = 1 (for x ≠ 0)
Law 5 — Negative Index
A negative exponent means "take the reciprocal." The negative sign does NOT make the answer negative — it moves the base from numerator to denominator (or vice versa).
- 2−3 = 1/2³ = 1/8 = 0.125
- 5−1 = 1/5 = 0.2
- 10−2 = 1/100 = 0.01
- x−4 = 1/x&sup4;
Law 6 — Fractional Index
A fractional index means "take a root." The denominator of the fraction is the root, and the numerator is the power. Always take the root first to keep numbers small.
- 91/2 = √9 = 3
- 81/3 = ³√8 = 2
- 82/3 = (³√8)² = 2² = 4 (root first, then power)
- 272/3 = (³√27)² = 3² = 9
- 163/4 = (&sup4;√16)³ = 2³ = 8
Combining Index Laws
In harder questions, you will need to apply multiple laws in sequence. Always state which law you are applying at each step.
Simplifying Surds
A surd is an irrational root that cannot be simplified to a whole number. To simplify a surd, extract the largest perfect square factor.
Perfect Squares You Must Know
1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36
7² = 49 8² = 64 9² = 81 10² = 100 11² = 121 12² = 144
The Method: Extracting Perfect Square Factors
- Find the largest perfect square that divides into the number under the root.
- Rewrite the surd as a product: √n = √(perfect square × remainder).
- Apply the rule √(a × b) = √a × √b.
- Simplify √(perfect square) to get a whole number.
Detailed Examples
The largest perfect square factor of 50 is 25 (since 50 = 25 × 2).
√50 = √(25 × 2) = √25 × √2 = 5√2
The largest perfect square factor of 72 is 36 (since 72 = 36 × 2).
√72 = √(36 × 2) = √36 × √2 = 6√2
The largest perfect square factor of 180 is 36 (since 180 = 36 × 5).
√180 = √(36 × 5) = √36 × √5 = 6√5
The largest perfect square factor of 288 is 144 (since 288 = 144 × 2).
√288 = √(144 × 2) = √144 × √2 = 12√2
Adding and Subtracting Surds
You can only add or subtract surds with the same number under the root (like surds). Think of it like algebra: you can add 3x + 5x = 8x, but you cannot simplify 3x + 5y.
- 3√5 + 7√5 = 10√5 (same root, add coefficients)
- 8√3 − 2√3 = 6√3
- 3√5 + 2√7 (different roots — leave as is)
√12 + √27 looks like unlike surds, but simplify each first:
√12 = √(4 × 3) = 2√3
√27 = √(9 × 3) = 3√3
Now they are like surds: 2√3 + 3√3 = 5√3
Multiplying Surds
- √3 × √5 = √15
- √6 × √10 = √60 = √(4 × 15) = 2√15
- 2√3 × 5√2 = (2 × 5) × √(3 × 2) = 10√6
- √5 × √5 = √25 = 5 (any surd times itself gives a whole number)
Expanding Brackets with Surds
Expand brackets with surds just like algebraic brackets. Use FOIL for two binomials.
= √3 × √3 + √3 × 4 = 3 + 4√3
= 2 × 3 + 2 × (−√5) + √5 × 3 + √5 × (−√5)
= 6 − 2√5 + 3√5 − 5
= 1 + √5
Rationalising the Denominator
In mathematics, we don't leave surds in the denominator. "Rationalising" means rewriting the fraction so the denominator is a rational number.
Method 1: Simple Denominator (√a)
If the denominator is a single surd, multiply top and bottom by that surd.
= (1 × √3) / (√3 × √3) = √3 / 3
= (6 × √2) / (√2 × √2) = 6√2 / 2 = 3√2
= (10 × √5) / (√5 × √5) = 10√5 / 5 = 2√5
Method 2: Conjugate Pairs (a + √b)
If the denominator contains a surd as part of a sum or difference (e.g. 2 + √3), multiply top and bottom by the conjugate: change the sign between the terms.
Standard Form (Scientific Notation)
Standard form is a way of writing very large or very small numbers compactly using powers of 10.
Converting to Standard Form
- Move the decimal point until you have a number between 1 and 10.
- Count how many places you moved it — that's the value of n.
- If you moved the decimal LEFT (large number), n is positive.
- If you moved the decimal RIGHT (small number), n is negative.
| Ordinary Number | Standard Form | Explanation |
|---|---|---|
| 4,500,000 | 4.5 × 10&sup6; | Decimal moves 6 places left |
| 320,000 | 3.2 × 10&sup5; | Decimal moves 5 places left |
| 0.0037 | 3.7 × 10−3 | Decimal moves 3 places right |
| 0.000 006 2 | 6.2 × 10−6 | Decimal moves 6 places right |
Calculating with Standard Form
(3 × 10&sup4;) × (2 × 10³) = (3 × 2) × 104+3 = 6 × 10&sup7;
(8 × 10&sup6;) ÷ (4 × 10²) = (8 ÷ 4) × 106−2 = 2 × 10&sup4;
(5 × 10³) × (4 × 10²) = 20 × 10&sup5; = 2 × 10&sup6; (adjust 20 to 2 × 10)
Worked Examples
These examples show the kind of multi-step reasoning expected at Grade 8. Notice how each step is justified.
√200 = √(100 × 2) = √100 × √2 = 10√2
Justification: I applied the surd multiplication rule √(ab) = √a × √b, which is valid for all positive a and b. The result 10√2 is fully simplified because 2 has no perfect square factors other than 1.
2³ × 2−5 = 23+(−5) = 2−2
Now I apply the negative index law: a−n = 1/an.
2−2 = 1/2² = 1/4 = 0.25
Justification: The multiplication law allows adding exponents when bases are identical. The negative index law converts a negative exponent to a positive one in the denominator.
Step 1: √8 = √(4 × 2) = 2√2, so 3√8 = 3 × 2√2 = 6√2
Step 2: √18 = √(9 × 2) = 3√2, so 2√18 = 2 × 3√2 = 6√2
Step 3: Now they are like surds: 6√2 + 6√2 = 12√2
163/4 = (4√16)³ = 2³ = 8
Check: 4√16 = 2 because 2&sup4; = 16. Then 2³ = 8. ✓
Step 2: Multiply top and bottom by the conjugate:
5(3 − √2) / [(3 + √2)(3 − √2)]
Step 3: Expand the denominator using difference of two squares:
(3)² − (√2)² = 9 − 2 = 7
Step 4: Expand the numerator:
5 × 3 − 5 × √2 = 15 − 5√2
Answer: (15 − 5√2) / 7
The denominator is now rational (7), so the expression is rationalised.
Standard form: 4.5 × 10−5
Part 2: (4.5 × 10−5) × (2 × 10³)
= (4.5 × 2) × 10−5+3
= 9 × 10−2
= 0.09
Practice Q&A
Attempt each question before revealing the model answer. Focus on justifying each step.
√98 = √(49 × 2) = √49 × √2 = 7√2
Method: Extracting perfect square factors using √(ab) = √a × √b.
I take the root first: 3√27 = 3 (because 3³ = 27).
Then raise to the power: 3² = 9.
Justification: Taking the root first keeps numbers small and avoids computing 27² = 729 first.
√12 = √(4 × 3) = 2√3, so 5√12 = 10√3
√48 = √(16 × 3) = 4√3, so 2√48 = 8√3
√75 = √(25 × 3) = 5√3
Combine: 10√3 − 8√3 + 5√3 = 7√3
Step 2: Divide: 3² ÷ 3&sup5; = 32−5 = 3−3 (division law)
Step 3: Convert: 3−3 = 1/3³ = 1/27 (negative index law)
8/√6 = (8 × √6) / (√6 × √6) = 8√6 / 6 = 4√6 / 3
(Simplified by dividing 8 and 6 by their common factor 2.)
The correct calculation: 2³ × 3² = 8 × 9 = 72 (evaluate each power separately).
The student's answer: 6&sup5; = 7776, which is far too large.
= √3 × √3 + √3 × (−2) + 5 × √3 + 5 × (−2)
= 3 − 2√3 + 5√3 − 10
= (3 − 10) + (−2√3 + 5√3)
= −7 + 3√3
Part 2: Time = Distance ÷ Speed
= (1.496 × 10&sup8;) ÷ (3 × 10&sup5;)
= (1.496 ÷ 3) × 108−5
= 0.4987 × 10³
= 4.987 × 10²
≈ 499 seconds ≈ 8 minutes 19 seconds
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
Add the exponents when multiplying with the same base.
Subtract the exponents when dividing with the same base.
Multiply the exponents when raising a power to a power.
A negative index means "take the reciprocal." It does NOT make the result negative.
The denominator is the root, the numerator is the power. Take the root first.
1/√a = √a/a