Powers, Roots and SurdsAQA GCSE Maths: Revision notes

Powers (or exponents) tell us how many times to multiply a number by itself. The number being multiplied is called the base, and the power is written as a small number above it.

• 2³ = 2 × 2 × 2 = 8 (read as '2 to the power of 3' or '2 cubed')
• 5² = 5 × 5 = 25 (read as '5 squared')
• 10⁴ = 10 × 10 × 10 × 10 = 10,000

Roots are the inverse (opposite) of powers. The square root (√) undoes squaring, and the cube root (∛) undoes cubing.

• √25 = 5 (because 5² = 25)
• ∛8 = 2 (because 2³ = 8)
• ⁴√81 = 3 (because 3⁴ = 81)

Recognising powers of common numbers helps speed up calculations:

Negative indices represent the reciprocal (one divided by the number). Instead of multiplying, you divide.

• a⁻ⁿ = 1/aⁿ
• 2⁻³ = 1/2³ = 1/8
• 5⁻² = 1/5² = 1/25

Fractional indices connect powers and roots. The denominator of the fraction tells you which root to take.

• a^(1/n) = ⁿ√a (the nth root of a)
• a^(m/n) = ⁿ√(aᵐ) or (ⁿ√a)ᵐ
• 27^(1/3) = ∛27 = 3
• 16^(3/4) = (⁴√16)³ = 2³ = 8
• 9^(-1/2) = 1/√9 = 1/3

Key points:

• Any number to the power 0 equals 1: a⁰ = 1
• A fractional index always means: root (denominator) then power (numerator)
• Negative fractional indices: apply both rules—take the root, then divide

The laws of indices are rules that simplify calculations with powers. They apply when the bases are the same.

Law 1: Multiplying powers — add the exponents

• aᵐ × aⁿ = aᵐ⁺ⁿ
• 2³ × 2⁵ = 2⁸
• x² × x⁷ = x⁹

Law 2: Dividing powers — subtract the exponents

• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• 3⁷ ÷ 3⁴ = 3³ = 27
• y⁶ ÷ y² = y⁴

Law 3: Power of a power — multiply the exponents

• (aᵐ)ⁿ = aᵐⁿ
• (2³)⁴ = 2¹²
• (x²)⁵ = x¹⁰

Law 4: Power of a product — apply the power to each factor

• (ab)ⁿ = aⁿbⁿ
• (2 × 3)² = 2² × 3² = 4 × 9 = 36
• (xy)³ = x³y³

Law 5: Power of a quotient — apply the power to numerator and denominator

• (a/b)ⁿ = aⁿ/bⁿ
• (2/3)³ = 2³/3³ = 8/27

A surd is a root of a number that cannot be simplified to a whole number or simple fraction. It must be left in root form to be exact.

• √2, √3, √5, √7 are surds (irrational numbers)
• √4 = 2 is NOT a surd (it simplifies to a whole number)
• √(1/4) = 1/2 is NOT a surd (it simplifies to a fraction)

Simplifying surds involves removing perfect square factors from under the root.

1. Find the largest perfect square that divides the number under the root
2. Split the surd into two: √(perfect square × other number)
3. Simplify: √(perfect square) × √(other number)

Examples:

• √12 = √(4 × 3) = √4 × √3 = 2√3
• √50 = √(25 × 2) = √25 × √2 = 5√2
• √72 = √(36 × 2) = √36 × √2 = 6√2

Rules for surds:

• √a × √b = √(ab)
• √a ÷ √b = √(a/b)
• You cannot simplify √a + √b unless a = b
• √2 + √3 cannot be simplified further
• 3√2 + 2√2 = 5√2 (like terms can be combined)

Rationalising the denominator means removing surds (roots) from the bottom of a fraction. This is required for exact answers on the GCSE.

For single surds in the denominator:

Multiply both numerator and denominator by the same surd to make the denominator rational (a whole number or fraction).

• 1/√2 = (1 × √2)/(√2 × √2) = √2/2
• 3/√5 = (3 × √5)/(√5 × √5) = 3√5/5
• 5/(2√3) = (5 × √3)/(2√3 × √3) = 5√3/(2 × 3) = 5√3/6

For denominators with surds and whole numbers (a + √b):

Multiply numerator and denominator by the conjugate (the same expression with the opposite sign).

• 1/(2 + √3) = [1 × (2 - √3)]/[(2 + √3) × (2 - √3)] = (2 - √3)/(4 - 3) = 2 - √3
• 4/(1 + √2) = [4 × (1 - √2)]/[(1 + √2) × (1 - √2)] = [4(1 - √2)]/(1 - 2) = [4(1 - √2)]/(-1) = -4 + 4√2

Key reminder: (a + b)(a - b) = a² - b², which is why the surd disappears from the denominator.

Estimating powers and roots means finding an approximate value, useful when exact calculation is difficult or unnecessary.

Strategy for estimating roots of any positive number:

1. Identify two perfect powers (or roots) that the number lies between
2. Decide whether the number is closer to the lower or upper value
3. Use known powers to narrow down your estimate

Example for √50:

• 7² = 49 and 8² = 64
• 50 is between 49 and 64, so √50 is between 7 and 8
• 50 is very close to 49, so √50 ≈ 7.1

Example for ∛200:

• 5³ = 125 and 6³ = 216
• 200 is between 125 and 216, so ∛200 is between 5 and 6
• 200 is closer to 216, so ∛200 ≈ 5.8 or 5.9

Estimating other powers:

• To estimate 2^7: use 2⁶ = 64, so 2⁷ = 128
• To estimate 5^4: use 5³ = 125, so 5⁴ = 625
• Use known facts about powers of 2, 3, 4, 5 to build estimates

Using powers of 10 for large numbers:

• 10³ = 1000, so the cube root of 1000 is 10
• Estimate √4000 by noting 60² = 3600 and 70² = 4900, so √4000 ≈ 63