Laws of Indices, Fractional & Negative Powers, and Simplifying Surds
Laws of indices provide the algebraic rules for multiplying, dividing, and raising powers with the same base, extending on the Higher Tier to negative indices (which take the reciprocal, x⁻ⁿ = 1/xⁿ) and fractional indices (where the denominator is the root and the numerator is the power, x^(m/n) = (ⁿ√x)ᵐ). Surds are irrational roots of integers (such as √2 or √5) that cannot be written as exact fractions and must be manipulated in exact form on Non-Calculator Paper 1. Examiners regularly test evaluating combined negative fractional powers of fractions, simplifying surd expressions, and rationalising two-term denominators using the difference of two squares conjugate.
- Basic Laws of Indices (Same Base)
- 1. Multiplication: xᵃ x xᵇ = x^(a+b) (add the powers). 2. Division: xᵃ ÷ xᵇ = x^(a-b) (subtract the powers). 3. Power of a power: (xᵃ)ᵇ = x^(ab) (multiply the powers). 4. Zero index: x⁰ = 1 (for any non-zero value of x). 5. Power of 1: x¹ = x.
- Negative Indices (Reciprocals)
- A negative index means 'take the reciprocal' (it does not make the number negative): x⁻ⁿ = 1 / xⁿ, and for a fraction, (a/b)⁻ⁿ = (b/a)ⁿ. For example, 5⁻² = 1 / 5² = 1/25, and (2/3)⁻³ = (3/2)³ = 27/8.
- Fractional Indices (Roots & Powers) (Higher Tier)
- A unit fraction index represents a root: x^(1/2) = √x (square root), x^(1/3) = ³√x (cube root), and x^(1/n) = ⁿ√x. For a general fractional index x^(m/n) = (ⁿ√x)ᵐ: Always take the n-th root first (to make the number smaller) before raising to the power m! For example, 64^(2/3) = (³√64)² = 4² = 16.
Quadratic Parabola Plotter, Roots, Turning Point & Quadratic Formula
Adjust coefficients a, b, and c in y = ax² + bx + c to see how the parabola, roots (x-intercepts), and turning point move on the graph.
y = x² - 2x - 3
(-2)² − 4(1)(-3) = 16 → 2 real roots (crosses x-axis twice)x = 1.00b in brackets when squaring: (-2)² = 4 (always positive!). Typing -2² into a calculator without brackets gives -4 and loses the accuracy mark.Key Concepts & Exam Definitions (6)
- Basic Laws of Indices (Same Base)
- 1. Multiplication: xᵃ x xᵇ = x^(a+b) (add the powers). 2. Division: xᵃ ÷ xᵇ = x^(a-b) (subtract the powers). 3. Power of a power: (xᵃ)ᵇ = x^(ab) (multiply the powers). 4. Zero index: x⁰ = 1 (for any non-zero value of x). 5. Power of 1: x¹ = x.
- Negative Indices (Reciprocals)
- A negative index means 'take the reciprocal' (it does not make the number negative): x⁻ⁿ = 1 / xⁿ, and for a fraction, (a/b)⁻ⁿ = (b/a)ⁿ. For example, 5⁻² = 1 / 5² = 1/25, and (2/3)⁻³ = (3/2)³ = 27/8.
- Fractional Indices (Roots & Powers) (Higher Tier)
- A unit fraction index represents a root: x^(1/2) = √x (square root), x^(1/3) = ³√x (cube root), and x^(1/n) = ⁿ√x. For a general fractional index x^(m/n) = (ⁿ√x)ᵐ: Always take the n-th root first (to make the number smaller) before raising to the power m! For example, 64^(2/3) = (³√64)² = 4² = 16.
- Changing to a Common Base Equations (Higher Tier)
- To solve index equations where the unknown x is in the power (e.g., 9^(x+1) = 27^(2x-1)), rewrite every number as a power of the same prime base (here, base 3: 9 = 3² and 27 = 3³): (3²)^(x+1) = (3³)^(2x-1) -> 3^(2x+2) = 3^(6x-3). Equate the powers: 2x + 2 = 6x - 3 -> 4x = 5 -> x = 1.25.
- Surd Rules & Simplifying Surds (Higher Tier)
- A surd is an irrational root that cannot be simplified to a rational number. Rules: 1. √a x √b = √(ab), and √a x √a = a. 2. √(a/b) = √a / √b. Warning: √(a + b) is never equal to √a + √b! To simplify √n, factor out the largest square factor k² so √(k²b) = k√b. To add or subtract surds, simplify each surd to a common root first (e.g., √50 - √18 = 5√2 - 3√2 = 2√2).
- Rationalising the Denominator (Higher Tier)
- Removing all surds from the denominator of a fraction to leave an integer denominator. Case 1 (Single surd a / √b): Multiply numerator and denominator by √b -> (a√b) / b. Case 2 (Two-term denominator c / (a + √b)): Multiply numerator and denominator by the conjugate (a - √b); the denominator expands via difference of two squares (a + √b)(a - √b) = a² - b.
Quick-Fire Flashcards (6)
Click any card to check the answer, then mark it as Known to track your progress.
Simplify completely: (4x⁵y⁻²)³ ÷ (2x³y⁻⁴).
Evaluate (16 / 81)^(-3/4) as an exact fraction without a calculator.
Simplify √75 + √48 - √12 in the form k√3.
Expand and simplify (4 - √3)² in the form a + b√3.
What do you multiply the top and bottom by to rationalise the denominator of 7 / (5 - 2√3)?
Express 8^(2x) x 4^(x-1) as a single power of 2.
Exam-Style Questions & Mark Schemes (4)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
Without using a calculator, work out the exact value of (125 / 64)^(-2/3) + 16^(1/2) - 9⁰. Show all steps of your working clearly. (Higher Tier) [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Apply the negative index to flip the fraction: (125 / 64)^(-2/3) = (64 / 125)^(2/3) [1 mark]. 2. Take the cube root of numerator and denominator: (³√64 / ³√125)² = (4 / 5)², then square to get 16 / 25 [1 mark]. 3. Evaluate the other two terms: 16^(1/2) = √16 = 4, and 9⁰ = 1, so 4 - 1 = 3 [1 mark]. 4. Add together: 3 + 16/25 = 3 and 16/25 (or 91/25 or 3.64) [1 mark].
- Q2
Solve the equation (25^(x)) / (5^(3x - 4)) = 125^(x + 2) by writing every term as a power of 5. (Higher Tier) [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Rewrite 25 as 5² and 125 as 5³: ((5²)^x) / (5^(3x - 4)) = (5³)^(x + 2) [1 mark]. 2. Apply the power of a power rule: (5^(2x)) / (5^(3x - 4)) = 5^(3x + 6) [1 mark]. 3. Apply the division index law on the left-hand side (subtract the powers carefully with brackets): 5^(2x - (3x - 4)) = 5^(-x + 4) = 5^(3x + 6) [1 mark]. 4. Equate the exponents and solve for x: -x + 4 = 3x + 6 -> -2 = 4x -> x = -0.5 (or -1/2) [1 mark].
- Q3
Rationalise the denominator and simplify fully: (14 + 2√5) / (3 - √5). Give your answer in the form a + b√5 where a and b are integers. (Higher Tier) [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Multiply both numerator and denominator by the conjugate of the denominator, (3 + √5): [(14 + 2√5)(3 + √5)] / [(3 - √5)(3 + √5)] [1 mark]. 2. Expand the numerator: (14 x 3) + 14√5 + 6√5 + 2(√5)² = 42 + 20√5 + 2(5) = 52 + 20√5 [1 mark]. 3. Expand the denominator using difference of two squares: 3² - (√5)² = 9 - 5 = 4 [1 mark]. 4. Divide both terms in the numerator by 4: (52 + 20√5) / 4 = 13 + 5√5 (so a = 13, b = 5) [1 mark].
- Q4
A rectangle has a length of (√80 + √20) cm and a width of (5√5 - √45) cm. Show that the area of the rectangle is an integer and find its perimeter in the form k√5 cm. (Higher Tier) [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Simplify the surds in the length into multiples of √5: √80 = √(16 x 5) = 4√5 and √20 = √(4 x 5) = 2√5, so Length = 4√5 + 2√5 = 6√5 cm [1 mark]. 2. Simplify the surds in the width into multiples of √5: √45 = √(9 x 5) = 3√5, so Width = 5√5 - 3√5 = 2√5 cm [1 mark]. 3. Calculate the area: Area = Length x Width = (6√5) x (2√5) = 12 x (√5 x √5) = 12 x 5 = 60 cm² (an integer) [1 mark]. 4. Calculate the perimeter: Perimeter = 2 x (Length + Width) = 2 x (6√5 + 2√5) = 2 x 8√5 = 16√5 cm [1 mark].
Common Exam Questions (4)
How do you evaluate a negative fractional power of a fraction, such as (8/27)^(-2/3), without a calculator?
Do it in three calm, separate steps. 1. Negative sign flips the fraction upside down (takes the reciprocal): (27/8)^(2/3). 2. Denominator of 3 takes the cube root of top and bottom: (³√27 / ³√8)² = (3/2)². 3. Numerator of 2 squares top and bottom: 3² / 2² = 9/4 (or 2.25).
What happens when you raise a term with a coefficient and a power inside brackets, like (3x⁴)³?
You must raise both the number coefficient and the algebraic variable to the outer power. Raise the number 3 to the power of 3 (3³ = 27) and multiply the indices on x (x^(4 x 3) = x¹²), giving 27x¹². A common error is multiplying 3 by 3 to get 9.
How do you simplify a surd like √72 into the form a√b?
Find the largest square number (from 4, 9, 16, 25, 36, 49, 64, 81, 100.) that divides into 72. Since 72 = 36 x 2, use the rule √(ab) = √a x √b: √72 = √36 x √2 = 6√2. You can only add or subtract surds once they have the exact same number under the root sign (like terms, e.g., 6√2 + 5√2 = 11√2).
How do you rationalise a two-term surd denominator like 10 / (3 - √5)? (Higher Tier)
Multiply both the numerator and the denominator by the conjugate of the denominator (keep the terms the same, but flip the sign in the middle from '-' to '+', so multiply top and bottom by (3 + √5)). By the difference of two squares, (3 - √5)(3 + √5) = 3² - (√5)² = 9 - 5 = 4, eliminating the surd from the denominator.
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