Multiplying the Notation

To multiply $2\cdot 10^{3}\cdot 3\cdot 10^{2}$, group the first parts together and the powers of ten together. The first parts give $2\cdot 3 = 6$. The powers of ten follow the product rule, so $10^{3}\cdot 10^{2} = 10^{5}$. The result is $6\cdot 10^{5}$, which is 600000.

To multiply two numbers in scientific notation, multiply the first parts and add the exponents of ten. Adding the exponents is the product rule for powers with the same base.

Sometimes the first parts multiply to 10 or more. If $4\cdot 5 = 20$, the result $20\cdot 10^{6}$ is not in scientific notation, because 20 is not less than 10. Move the decimal point one more place to the left and add 1 to the exponent: $2\cdot 10^{7}$. Always check the first part at the end.