Powers and roots - formulas

The power of natural exponent \[a^n=\underbrace{a\cdot a\cdot a\cdot...\cdot a}_{n \text{ times}}\]

The power of rational exponent \[ a^{-n}=\frac{1}{a^n}\quad (\text{for }a\ne 0)\\[16pt] a^{\tfrac{1}{n}}=\sqrt[n]{a}\quad (\text{for }a\ge 0)\\[16pt] a^{\tfrac{k}{n}}=\sqrt[n]{a^k}\quad (\text{for }a\ge 0)\\[16pt] a^{-\tfrac{k}{n}}=\frac{1}{\sqrt[n]{a^k}}\quad (\text{for }a\gt 0)\\[16pt] \]

Powers - formulas \[ a^m\cdot a^n=a^{m+n}\\[16pt] \frac{a^m}{a^n}=a^{m-n}\\[16pt] a^n\cdot b^n=(a\cdot b)^n\\[16pt] \frac{a^n}{b^n}=\left (\frac{a}{b}\right )^n\\[16pt] \left(a^m \right)^n=a^{m\cdot n} \]

Roots - formulas \[ \sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}\\[16pt] \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}} \] Actions on more complex roots is usually done by transforming them into powers. \[ \sqrt[n]{a}=a^{\tfrac{1}{n}}\\[16pt] \sqrt[n]{a}\cdot \sqrt[m]{a}=a^{\tfrac{1}{n}}\cdot a^{\tfrac{1}{m}}=a^{\tfrac{1}{n}+\tfrac{1}{m}}\\[16pt] \frac{\sqrt[n]{a}}{\sqrt[m]{a}} =\frac{a^{\tfrac{1}{n}}}{a^{\tfrac{1}{m}}} =a^{\tfrac{1}{n}-\tfrac{1}{m}}\\[16pt] \]

Another formulas \[ a^0=1\quad (\text{for }a\ne 0)\\[16pt] \sqrt{a^2}=|a| \]