Part
1: How Do we Multiply numbers in Scientific Notation?

Scientific Notation is based on powers of the base number 10.

The number 123,000,000,000 in scientific notation is written
as :

The first number 1.23 is called the coefficient. It must be greater
than or equal to 1 and less than 10.

The second
number is called the base . It must always be 10 in scientific notation.
The base number 10 is always written in exponent form. In the number 1.23 x 10^{11}
the number 11 is referred to as the exponent or power of ten.

Rules
for Multiplication in Scientific Notation:

1) Multiply
the coefficients

2) Add the exponents (base 10 remains)

Example 1: (3 x 10^{4})(2x 10^{5}) = 6
x 10^{9}

What happens if the coefficient
is more than 10 when using scientific notation?

Example
2: (5 x 10 ^{3}) (6x 10^{3}) = 30. x 10^{6 }

While the value is correct it is not correctly written in scientific notation,
since the coefficient is not between 1 and 10. We then must move the decimal point
over to the left until the coefficient is between 1 and 10. For each place we
move the decimal over the exponent will be raised 1 power of ten.

30.x10^{6}
= 3.0 x 10^{7}^{ }in scientific notation.

Example
3:

(2.2 x 10^{ 4})(7.1x 10^{ 5}) = 15.62
x 10^{ 9} = 1.562 x 10^{ 10}

Example
4:

(7 x 10^{4})(5 x 10^{6})(3 x 10^{2})
= 105. x 10^{ 12} --now the decimal must be moved two places over and
the exponent is raised by 2. Therefore the value in scientific notation is: 1.05
x 10^{ 14}

Now
Try these:

(write
your answers in the form of coefficientx10^exponent) If your answer is 3.5 x 10^{
3 } you should type 3.5x10^3 in the box then click the submit
button).

What happens when the exponent(s) are negative?

We
still add the exponents, but use the rules of addition of signed numbers.

Example
5: (3 x 10 ^{-3}) (3x 10^{-3}) = 9. x 10^{-6 }

Example 6: (2 x 10 ^{-3}) (3x 10^{8})
= 6. x 10^{5}

Now
Try these:

(write
your answers in the form of coefficientx10^exponent) If your answer is 3.5 x 10^{
3 } you should type 3.5x10^3 in the box then click the submit
button). Multiply the
following: