Laws of indices
- aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ.
- a⁰ = 1, and a negative power means a reciprocal: a⁻ⁿ = 1/aⁿ.
- A fractional power means a root: a^(1/n) is the nth root of a, and a^(m/n) = (nth root of a)ᵐ. Take the root first to keep numbers small.
- Examples: 8^(2/3) = (∛8)² = 2² = 4. 27^(−1/3) = 1/∛27 = 1/3. (4/9)^(−1/2) = (9/4)^(1/2) = 3/2.
Solving index equations
- Write both sides with the same base, then compare powers. 2ˣ = 32 = 2⁵, so x = 5.
- 9ˣ = 27 means (3²)ˣ = 3³, so 2x = 3 and x = 3/2.
Simplifying surds
- A surd is a root that can't be simplified to a whole number, such as √2.
- √(ab) = √a × √b and √(a/b) = √a ÷ √b. Look for square factors: √72 = √36 × √2 = 6√2.
- Add like surds: √12 + √27 = 2√3 + 3√3 = 5√3.
- Expand brackets as normal, remembering √a × √a = a: (1 + √2)² = 1 + 2√2 + 2 = 3 + 2√2, and (2 + √3)(2 − √3) = 4 − 3 = 1.
Rationalising the denominator
- To remove √a from the denominator, multiply top and bottom by √a: 6/√3 = 6√3/3 = 2√3.
- For a denominator a + √b, multiply top and bottom by a − √b (and the other way round), using the difference of two squares.
- Example: 4/(3 − √5) = 4(3 + √5)/((3 − √5)(3 + √5)) = 4(3 + √5)/(9 − 5) = 3 + √5.
Key terms
- Index
- The power a number is raised to.
- Reciprocal
- One divided by a number.
- Fractional index
- A power such as 1/n, meaning the nth root.
- Surd
- A root that can't be simplified to a whole number.
- Rationalising the denominator
- Rewriting a fraction so there is no surd in the denominator.
- Like surds
- Surds with the same number under the root, which can be added.