## Intermediate Algebra (12th Edition)

$\text{Scientific notation: } 3.0\times10^{-4} \\\text{Standard form: } 0.0003$
A number in scientific notation takes the form $a\times10^n$ where $1\le a\lt10$ and $n$ is an integer. Hence, the given expression, $\dfrac{2,500,000\times0.00003}{0.05\times5,000,000} ,$ is equivalent to \begin{array}{l}\require{cancel} \dfrac{(2.5\times10^6)(3.0\times10^{-5})}{(5.0\times10^{-2})(5.0\times10^{6})} .\end{array} Using the law of exponents which states that $a^x\cdot a^y=a^{x+y},$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{(2.5)(3.0)\times10^{6+(-5)}}{(5.0)(5.0)\times10^{-2+6}} \\\\= \dfrac{7.5\times10^{1}}{25\times10^{4}} .\end{array} Using the law of exponents which states that $\dfrac{a^x}{a^y}=a^{x-y},$ the expression above simplifies to \begin{array}{l}\require{cancel} 7.5\div25\times10^{1-4} \\\\= 0.3\times10^{-3} \\\\= 0.3\times10^{-3} .\end{array} Hence, the simplified form is \begin{array}{l}\require{cancel} \text{Scientific notation: } 3.0\times10^{-4} \\\text{Standard form: } 0.0003 .\end{array}