Number: Indices, Surds and Standard Form

At GCSE you extend your knowledge of number to include fractional and negative indices, surds, standard form operations, and limits of accuracy including upper and lower bounds.

---

Laws of Indices

Law	Rule	Example
Multiply same base	aᵐ × aⁿ = aᵐ⁺ⁿ	3⁴ × 3⁵ = 3⁹
Divide same base	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁷ ÷ 5³ = 5⁴
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(2³)⁴ = 2¹²
Zero index	a⁰ = 1	7⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ	4⁻² = 1/16
Fractional index (root)	a^(1/n) = nth root of a	27^(1/3) = 3
Fractional index (power and root)	a^(m/n) = (nth root of a)ᵐ	8^(2/3) = (cube root of 8)² = 4

Example: Evaluate 16^(3/4). 16^(1/4) = 2 (fourth root); 2³ = 8

Example: Simplify (2x³y)⁴. = 2⁴ × x¹² × y⁴ = 16x¹²y⁴

---

Standard Form

Standard form: A × 10ⁿ where 1 ≤ A < 10 and n is an integer.

Calculations in Standard Form

Multiplying: multiply the A values and add the indices. (3 × 10⁵) × (4 × 10³) = 12 × 10⁸ = 1.2 × 10⁹

Dividing: divide the A values and subtract the indices. (9 × 10⁷) ÷ (3 × 10⁴) = 3 × 10³

Adding/Subtracting: convert to the same power of 10 first, or convert to ordinary numbers, calculate, then convert back. (4 × 10⁴) + (6 × 10³) = 40,000 + 6,000 = 46,000 = 4.6 × 10⁴

---

Surds [H]

A surd is an irrational root that cannot be written as a fraction — e.g. √2, √3, √5. It is exact, unlike a decimal approximation.

Simplifying Surds

Look for the largest perfect-square factor. √72 = √(36 × 2) = 6√2

Adding and Subtracting Surds

Only like surds can be added or subtracted. 3√5 + 7√5 = 10√5 5√3 − 2√3 = 3√3

Multiplying Surds

√a × √b = √(ab); (a√b)² = a²b

Example: (3 + √2)(4 − √2) = 12 − 3√2 + 4√2 − 2 = 10 + √2

Rationalising the Denominator [H]

Remove the surd from the denominator.

Monomial denominator: multiply by √a/√a. 5/√3 = 5√3/3

Binomial denominator: multiply by the conjugate. 3/(2 + √5) × (2 − √5)/(2 − √5) = 3(2 − √5)/(4 − 5) = 3(2 − √5)/(−1) = −3(2 − √5) = −6 + 3√5

---

Limits of Accuracy

When a measurement is rounded, the true value lies within a range.

If x = 7.4 (rounded to 1 d.p.): 7.35 ≤ x < 7.45

• Lower bound: subtract half the degree of accuracy from the rounded value.
• Upper bound: add half the degree of accuracy (the upper bound is NOT included — strict inequality).

Error Intervals

Written as: lower bound ≤ x < upper bound

Example: A length L is measured as 12 cm to the nearest cm. Error interval: 11.5 ≤ L < 12.5

Calculations with Bounds [H]

• Maximum value of a sum → add upper bounds.
• Minimum value of a sum → add lower bounds.
• Maximum value of a product → multiply upper bounds.
• Maximum value of a division → divide the upper bound of the numerator by the lower bound of the denominator.

Example: p = 5.6 (1 d.p.), q = 3.2 (1 d.p.). Find the upper bound of p/q. Upper bound = 5.65/3.15 = 1.794…

---

Product Rule for Counting [H]

If there are m ways to do the first thing and n ways to do the second, there are m × n ways to do both.

Example: A restaurant offers 4 starters, 6 mains and 3 desserts. How many different 3-course meals? 4 × 6 × 3 = 72 combinations

---

Recurring Decimals as Fractions

Let x equal the recurring decimal, then multiply by a power of 10 to shift the recurring part, and subtract.

Example: Express 0.27̄ (0.2777…) as a fraction. x = 0.2777… → 10x = 2.777… → 100x = 27.777… 100x − 10x = 27.777… − 2.777… = 25 90x = 25 → x = 25/90 = 5/18

---

Practice

1. Evaluate: (a) 64^(2/3); (b) 125^(−1/3); (c) 32^(0.6).
2. Simplify: (3a²b³)² × 2a⁻¹b.
3. Express 5.4 × 10⁻³ ÷ 2 × 10⁻⁷ in standard form.
4. Simplify √200 − 3√8 + √18.
5. [H] Rationalise the denominator: (4 + √3)/(2 − √3).
6. A distance d is measured as 28.5 m to 3 significant figures. Write the error interval for d.
7. [H] p = 4.7 (1 d.p.) and q = 2.1 (1 d.p.). Find the lower bound of p − q.
8. Express 0.13̄6̄ (0.136136…) as a fraction in its simplest form.
9. A school has 5 routes from home and 3 routes from the bus stop to school. How many different journeys are possible?
10. [H] Solve 2^(x+1) = 32. (Hint: express both sides as powers of 2.)