# How do you express 8,400,000 in scientific notation?

Mar 9, 2017

$8.4 \times {10}^{6}$

#### Explanation:

Expressing a number in scientific notation means you will adjust the given number to result a more compact form of the number that may include rounding it off to match the precision of other numbers given.

Scientific notation is a compact way of writing numbers to reduce complicated computations using very large or very small numbers and to reduce errors.

In our example: $8 , 400 , 000 = 8.4 \times {10}^{6}$ in scientific notation.

To get this result we divided the given larger number repeatedly by 10, dropping a zero each time. While we did that, we added another $1$ to the exponent of the $10$ multiplier that is at the end of every number in scientific notation. It looks like this:

$8 , 400 , 000 = 8400000 \times {10}^{0}$ because we know ${10}^{0} = 1$
$8 , 400 , 000 = 840000 \times {10}^{1}$ drop one $0$ and exponent is $1$.
$8 , 400 , 000 = 84000 \times {10}^{2}$ drop two $0$ and exponent is $2$.
$8 , 400 , 000 = 8400 \times {10}^{3}$ drop three $0$ and exponent is $3$.
$8 , 400 , 000 = 840 \times {10}^{4}$ drop four $0$ and exponent is $4$.
$8 , 400 , 000 = 84 \times {10}^{5}$ drop five $0$ and exponent is $5$.

Having run out of zeros, we could stop here and not be numerically incorrect, but there is a definition of scientific notation called "normalized" that requires the first digit to reside between $0$ and $9$.

So to become normalized, we must divide our number again by $10$ so the first digit becomes $8$.
Remember to add to the exponent.

Then: $8 , 400 , 000 = 8.4 \times {10}^{6}$

If you understand that method the easy way follows:

Place a dot (decimal point) at the right hand side of the original number of $8400000$. Notice we removed the commas.

Now move the decimal point to the left counting each time you pass a digit $8.4 .0 .0 .0 .0 .0 .$

If you count each $m o v e$ correctly you should reach $6$ when you land between $8$ and $4$.

Then: $8 , 400 , 000 = 8.4 \times {10}^{6}$