Powers & Roots¶
- Base
- Exponent
- \(x^n = \prod \limits_{i=1}^n x\)
- Negative number raised to even power: positive
- Negative number raised to odd power: negative
- Power laws
- \(a^m \times a^n = a^{m+n}\)
- \(a^m \times b^m = (a b)^{m}\)
- \(a^0 = 1\)
- \((a^m)^n = a^{mn}\)
- \(x^{-n}=\dfrac{1}{x^n}\)
- \((a^m b^n)^o = (a^{mo} b^{no})\)
- \(\sqrt[n]{x} = x^{1/n}\)
- \(x^{m/n} = (x^m)^{1/n} = (x^{1/n})^m\)
- \(x^m=x^n \iff m=n \quad (x \not \in [-1, 0, 1])\)
- Roots
- Odd root of negative number will be negative
- Odd root of positive number will be positive
- We cannot find even root of negative number
- Rationalizing
- Multiply numerator and denominator by conjugate of denominator
Units Digit¶
- Look for repeating pattern
- Figure out where pattern will be at desired power
- The units digit of any product will be influenced only by the units digits of the 2 factors
Eg: What is the unit’s digit of \(57^{123}\)