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Surds and indices

The laws of indices — and why negative and fractional powers have to mean what they mean — plus how to simplify surds and rationalise a denominator.

Pure · Algebra & functions

Surds and indices

The laws of indices — and why negative and fractional powers have to mean what they mean — plus how to simplify surds and rationalise a denominator.

Why it works

The three index laws are bookkeeping for repeated multiplication: am×an=am+n,am÷an=am−n,(am)n=amn.a^m \times a^n = a^{m+n}, \qquad a^m \div a^n = a^{m-n}, \qquad (a^m)^n = a^{mn}. Count the factors and they're obvious: a2×a3=(a a)(a a a)=a5a^2 \times a^3 = (a\,a)(a\,a\,a) = a^5.

The interesting part is what happens when the power isn't a counting number. You don't get to choose what a0a^0, a−na^{-n} or a1/2a^{1/2} mean — they're forced, because the laws above have to keep working.
  • Zero. am÷am=am−m=a0a^m \div a^m = a^{m-m} = a^0. But anything over itself is 11, so
a0=1a^0 = 1 (for a≠0a \ne 0).
  • Negative. a−n×an=a0=1a^{-n} \times a^{n} = a^{0} = 1, so a−na^{-n} is whatever you
multiply ana^n by to get 11 — its reciprocal: a−n=1ana^{-n} = \dfrac{1}{a^n}. A negative power means "one over", not "make it negative": 2−3=182^{-3} = \frac18, not −8-8.
  • Fractional. a1/2×a1/2=a1=aa^{1/2} \times a^{1/2} = a^{1} = a, so a1/2a^{1/2} is the thing
that squares to aa — the square root. In general the bottom is the root, the top is the power: am/n=(an)m=amn.a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}. Take the root first and the numbers stay small: 163/4=(164)3=23=816^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8.

Surds are roots that don't come out whole, like 2\sqrt2 — you keep them exact rather than rounding. The one rule that does the work is ab=a b.\sqrt{ab} = \sqrt{a}\,\sqrt{b}. To simplify a surd, pull out the largest square factor: 50=25×2=25 2=52\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25}\,\sqrt2 = 5\sqrt2. The warning that goes with this: the rule is for products, not sums — a+b≠a+b\sqrt{a+b} \ne \sqrt a + \sqrt b (try a=b=2a=b=2: 4=2\sqrt4 = 2, but 2+2≈2.83\sqrt2 + \sqrt2 \approx 2.83).

Rationalising means clearing a surd out of the denominator. A single surd: multiply top and bottom by it, 15=55\dfrac{1}{\sqrt5} = \dfrac{\sqrt5}{5}. A sum like a+ba + \sqrt b on the bottom: multiply by its conjugate a−ba - \sqrt b, because (a+b)(a−b)=a2−b(a+\sqrt b)(a-\sqrt b) = a^2 - b — a difference of two squares, which kills the root.

Expanding and factorising are the other half of handling algebraic expressions. Expanding multiplies brackets out and collects like terms — (2x−3)(x+4)=2x2+5x−12(2x - 3)(x + 4) = 2x^2 + 5x - 12. Factorising reverses it: always take out any common factor first, then factorise what's left, e.g. 2x3−8x=2x(x2−4)=2x(x−2)(x+2)2x^3 - 8x = 2x(x^2 - 4) = 2x(x - 2)(x + 2) (the bracket is a difference of two squares).