Surds and Indices.

Notes

Key idea

A surd is an irrational root left in exact form, such as √2 or 3√5. Exact answers keep the surd; a decimal is only an approximation.

Surd rules

• √a × √b = √(ab)
• √a ÷ √b = √(a/b)
• (√a)² = a
• To simplify, pull out the largest square factor: √72 = √(36 × 2) = 6√2
• Only "like" surds add or subtract: 3√2 + 5√2 = 8√2, but √2 + √3 cannot be combined.

Rationalising the denominator

• Single surd: multiply top and bottom by the surd. 6/√3 = 6√3/3 = 2√3
• Binomial: multiply top and bottom by the conjugate (change the sign between the terms).
  1/(3 + √2) = (3 − √2) / ((3 + √2)(3 − √2)) = (3 − √2)/7
• Useful identity: (a + √b)(a − √b) = a² − b, so the surd always disappears.

Index laws

Worked examples

1. Evaluate 8^(−2/3). Root first, then power, then reciprocal: ∛8 = 2, 2² = 4, so the answer is 1/4.
2. Expand (2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3.
3. Solve 9^x = 27. Write both as powers of 3: 3^(2x) = 3³, so 2x = 3 and x = 3/2.

Common mistakes

• Writing √(a + b) = √a + √b. This is false.
• Forgetting that a negative power means a reciprocal, not a negative number.
• Rationalising with the wrong sign in the conjugate.
• Leaving a surd in the denominator of a final answer.

Exam tips

• "Show that" and "exact value" questions need surd or fraction answers, never decimals.
• For equations with powers, write every term with the same base first.
• For fractional powers, do the root first. It keeps the numbers small.