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.

Rational and irrational

Ends or repeats: rational. Neither: irrational.

Watch out: a repeating decimal is a fraction over 9, 99, …

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 $1$.

Polynomial arithmetic

• Add: combine like terms.
• Multiply: every term by every term; degrees add.

Watch out: $x$-terms add; constants multiply.