Exponents obey a handful of rules that simplify many calculations. The product rule states that when powers share the same base, their exponents are added: \(a^m \cdot a^n = a^{m+n}\). The power rule says that raising a power to another power multiplies the exponents: \((a^m)^n = a^{mn}\). The quotient rule says that dividing powers with the same base subtracts the exponents: \(\frac{a^m}{a^n} = a^{m-n}\) (with \(a \neq 0\)). From the quotient rule, any nonzero number raised to the zero power equals \(1\), because \(a^n / a^n = a^{n-n} = a^0 = 1\). A negative exponent, meanwhile, signals a reciprocal: \(a^{-n} = \frac{1}{a^n}\), so \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).
Radicals are closely tied to fractional exponents. The \(n\)th root of \(a\) can be written as \(a^{1/n}\), and more generally \(\sqrt[n]{a^m} = a^{m/n}\). This connection lets you convert freely between radical and exponential form. To simplify a radical, factor the radicand to expose perfect powers, as in \(\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\). When a radical appears in a denominator, it is standard practice to rationalize the denominator by multiplying both numerator and denominator by the radical, so that \(\frac{1}{\sqrt{3}}\) becomes \(\frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}\).
A rational expression is a fraction whose numerator and denominator are both polynomials, such as \(\frac{x^2 + 1}{x - 3}\). Such an expression is undefined wherever the denominator equals zero, so the first step in working with rational expressions is to identify and state the restrictions on the variable. To simplify, factor the numerator and denominator completely and cancel any common factors. Two rational expressions are multiplied by multiplying numerators together and denominators together, and they are divided by multiplying the first expression by the reciprocal of the second. To add or subtract rational expressions with different denominators, find the least common denominator, rewrite each fraction with that LCD, and then combine the numerators before simplifying the result.