# Exponent Rules For logarithm rules see [[Logarithm Rules]]. For square roots, see [[Square Root]]. Basic properties: $ \begin{gathered} x^ax^b = x^{a+b} \\ x^0 = 1 \end{gathered} $ Derived properties (see [[#Deriving Exponent Properties From Basic Properties]] for proof): $ \begin{gathered} \left(x^a\right)^b = x^{ab} \\ x^{\frac 1 a} = \root a \of x \\ x^{-a} = \frac{1}{x^a} \end{gathered} $ ## Exponent Change of Base Formula ![[Logarithm Rules#^hmy4h0]] So we can can say that $j^k = c^{\log_c{(j^k)}} = c^{\log_c(j)k}$ ^k5kjan ## Rules that are sometimes true $ \begin{gathered} x^a y^a = (xy)^a \\ \sqrt x \sqrt y = \sqrt {xy} \end{gathered} $ True when at least one of $x$, $y$ are positive, or when exponent $a$ is an integer. See [[Misconception About Multiplying Square Roots]]. $ \sqrt \frac{x}{y} = \frac{\sqrt x}{\sqrt y} $ The above is true when $y$ is positive, or when both $x$ and $y$ are negative. See [[Misconception About Roots and Fractions]]. ## Deriving Exponent Properties From Basic Properties $ \left(x^a\right)^b = \prod_{i=1}^b x^a = x^{\sum_{i=1}^b a} = x^{ab} $ $ \begin {gathered} \left(x^{\frac 1 a}\right)^a = \prod_{i=1}^a x^{\frac 1 a} = x^{\sum_{i=1}^a \frac 1 a} = x^1 = x \\ \left(x^{\frac 1 a}\right)^a = x \\ x^{\frac 1 a} = \root a \of x \end{gathered} $ $ \begin{gathered} x^{-a} x^{a} = x^{-a+a} = x^0 = 1 \\ x^{-a} x^{a} = 1 \\ x^{-a} = \frac{1}{x^a} \end{gathered} $