Revision sheet

Rational exponents

• $b^{\frac{1}{n}} = \sqrt[n]{b}$; $b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m$.
• $b^{-r} = \dfrac{1}{b^{r}}$: the value stays positive.
• A factor $b$ per unit means a factor $b^{\frac12}$ per half-unit.

Radical equations

Isolate → square (or cube) → solve → check.

Watch out: squaring can add extraneous solutions.

The number i

$i^2 = -1$; $\sqrt{-a} = i\sqrt{a}$; powers of $i$ repeat every 4.

Watch out: $\sqrt{-16}$ is $4i$, not $-4$.

Complex arithmetic

Add parts separately; multiply out and use $i^2 = -1$; conjugates give a real product.

Watch out: $i^2$ is $-1$, not $1$.

Complex solutions

Discriminant < 0 → $x = p \pm qi$ (conjugates).

Watch out: divide the whole numerator by $2a$.

Complex zeros

• $i^2 = -1$, so $\sqrt{-k^2} = ki$.
• $b^2 - 4ac < 0$: no real zeros, two complex ones.
• Real coefficients: $p + qi$ and $p - qi$ come together.

Watch out: the conjugate changes only the sign of the $i$ part.