Exponents & roots

An exponent tells how many times to multiply a number by itself: $2^3=2\cdot 2\cdot 2$. A fractional exponent means a root: $x^{1/2}=\sqrt{x}$ and $x^{2/3}=\sqrt[3]{x^2}$. Negative exponents flip the base: $x^{-n}=\frac{1}{x^n}$. To compare powers with different exponents, rewrite them so the exponents match: $2^{300}=(2^3{)}^{100}=8^{100}$ and $3^{200}=(3^2{)}^{100}=9^{100}$. Since $9>8$, $3^{200}$ is larger. To simplify radicals, pull out perfect square factors: $\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}$. Then combine like terms: $5\sqrt{2}+3\sqrt{2}=8\sqrt{2}$. For nested radicals, simplify from the inside out.