Surds & Indices

Basic rules governing powers.

• $a^m \times a^n = a^{m+n}$
• $a^m / a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^{-n} = 1/a^n$
• $a^0 = 1$ (for $a \neq 0$)

Solve infinite nested roots in 2 seconds.

• Addition: $\sqrt{x + \sqrt{x + ...}} =$ Large Factor of x. (where diff of factors is 1).
• Subtraction: $\sqrt{x - \sqrt{x - ...}} =$ Small Factor of x.
• Multiplication: $\sqrt{x \sqrt{x ...}} = x$.

To compare $\sqrt[a]{x}$ and $\sqrt[b]{y}$:
1. Take LCM of power indices ($a, b$).
2. Raise numbers to the power of LCM.
3. Compare resulting integers.

Eliminate roots from denominator by multiplying with Conjugate.

• Conjugate of $\sqrt{a} + \sqrt{b}$ is $\sqrt{a} - \sqrt{b}$.
• Formula: $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b$.

To find $\sqrt{A \pm 2\sqrt{B}}$, find two numbers $x, y$ such that $x+y=A$ and $xy=B$. Then answer is $\sqrt{x} \pm \sqrt{y}$.

• Golden Rule: Ensure coefficient of inner root is 2. If not, multiply/divide or adjust.