Surds & Indices

Everything in this topic comes out of six laws. Learn them once and the questions become one-line.

• a^m × a^n = a^(m+n) — same base, add the powers.
• a^m ÷ a^n = a^(m−n) — same base, subtract.
• (a^m)^n = a^(mn) — power of a power, multiply.
• a^0 = 1 for every a except 0.
• a^(−n) = 1/a^n.
• ⁿ√a = a^(1/n), so √a = a^(1/2).

Three question types follow:

1. Simplify an expression in one base — just apply the laws.
2. Solve for x in a^(x−k) = N — write N as a power of a, equate the indices, then undo the −k. Forgetting that last step is a favourite trap.
3. Rationalise a denominator — multiply top and bottom by the conjugate: for 1/(√a + √b) use √a − √b, which turns the denominator into the plain number a − b.

The single most common mistake is adding when you should multiply: a^m × a^n adds the indices, but (a^m)^n multiplies them. Read whether the powers are side by side or stacked.

Comparing surds of different orders (√2 vs ∛3): raise each to the LCM of the orders — here the 6th power — and compare plain integers.

a^m × a^n = a^(m+n) · (a^m)^n = a^(mn) · a^0 = 1 · a^(−n) = 1/a^n · conjugate of √a + √b is √a − √b

If 3^(x−2) = 243, find x. → Write 243 as a power of 3: 3, 9, 27, 81, 243 — that is 3^5. So x − 2 = 5 and x = 7. Sanity check: 3^(7−2) = 3^5 = 243. Stopping at x = 5 forgets to undo the −2, and that is exactly the distractor the paper prints.