2. The Formula Sheet & Every Tool You Need

You get the formula sheet, but the elite items test whether you can reach for the right tool fast and combine tools.

Exponents & radicals

• \(a^{m/n} = \left(\sqrt[n]{a}\right)^m\); \(a^{-n} = \dfrac{1}{a^n}\); rationalize by multiplying by \(\dfrac{\sqrt{b}}{\sqrt{b}}\).

Geometry

• Sphere: volume \(\tfrac43\pi r^3\), surface area \(4\pi r^2\).
• Equilateral triangle area: \(\dfrac{\sqrt3}{4}s^2\); 30-60-90 sides are \(x : x\sqrt3 : 2x\); 45-45-90 are \(x : x : x\sqrt2\).
• Inscribed angle in a semicircle is \(90^\circ\).

Algebra & sequences

• Quadratic formula \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\); vertex form \(y=a(x-h)^2+k\) has vertex \((h,k)\).
• Arithmetic: \(a_n=a_1+(n-1)d\). Geometric: \(a_n=a_1 r^{\,n-1}\).
• Inverse function: swap \(x\) and \(y\), then solve for \(y\).

Counting & probability

• Permutations \({}_nP_r=\dfrac{n!}{(n-r)!}\); combinations \({}_nC_r=\dfrac{n!}{r!(n-r)!}\).
• Addition rule: \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\). Conditional: \(P(B\mid A)=\dfrac{P(A\text{ and }B)}{P(A)}\).

Growth & decay

• Compound: multiply by \((1+\tfrac{r}{n})\) per period; half-life: multiply by \(\tfrac12\) each half-life.