# How do you simplify and write (3.11times10^3)(1.01times10^13) in standard form?

Apr 15, 2017

$3.1411 \times {10}^{16} = 31 , 411 , 000 , 000 , 000 , 000.$

#### Explanation:

Consider the product: $\text{ } 3 {x}^{4} \times 5 {x}^{7}$

This can also be written as $\text{ } \textcolor{b l u e}{3 \times 5} \times \textcolor{red}{{x}^{4} \times {x}^{7}}$

Multiply the numbers and add the indices of like bases.

$3 {x}^{4} \times 5 {x}^{7} = \textcolor{b l u e}{15} \textcolor{red}{{x}^{11}}$

In the same way: $\text{ } 3.11 \times {10}^{3} \times 1.01 \times {10}^{13}$

Re-arrange to give: $\text{ } \textcolor{b l u e}{3.11 \times 1.01} \times \textcolor{red}{{10}^{3} \times {10}^{13}}$

Multiply the numbers and add the indices of like bases.

$= \textcolor{b l u e}{3.1411} \times \textcolor{red}{{10}^{16}}$

In some cases the number ends up as a number of 10 or more and has to be adjusted to be in scientific notation.

In decimal notation $3.1411 \times {10}^{16} = 31 , 411 , 000 , 000 , 000 , 000.$

$3.2 \times {10}^{8} \times 6.5 \times {10}^{11} = 20.8 \times {10}^{19} = 2.08 \times {10}^{20}$