Multiplying Numbers Written in Scientific Notation

Date: 06/27/2004 at 13:06:58
From: Tommy
Subject: pre-algebra

I don't understand how to do this problem and reading about scientific
notation has confused me even more.  Please explain in full detail if
you have time.

(3 x 10^3)(1.5 x 10^5) 

If the numbers are already in scientific notation then how do you
multiply them?  What are the steps?

Date: 06/27/2004 at 20:34:03
From: Doctor Riz
Subject: Re: pre-algebra

Hi Tommy -

Thanks for writing to Dr. Math.  I'm not sure if you are confused
about scientific notation itself or just how to multiply two numbers
that are written in that form, so I'll answer both questions.

Scientific notation is a way of writing very large or very small
numbers in a compact form.  The number in front is always written with
one digit before the decimal (ie 3.28, 4.0 and so on) and the power of
10 is the key to telling us how far away from that position the
decimal is in the expanded form of the number.

As you may know, multiplying a number by 10 (or 10^1) has the effect
of moving the decimal one place to the right:

  4.56 * 10 = 45.6

Multiplying by 100 (or 10^2) moves the decimal two places to the right
and by 1000 (or 10^3) moves it three places to the right:

  4.56 *  100 = 456.  =  456
  4.56 * 1000 = 4560. = 4560

So a number written in scientific notation shows you where the decimal
starts and tells you how far to move it to get the expanded number:

  4.56 x 10^5 = 456000. = 456,000 since the decimal moves 5 places to
  the right

Similarly, dividing by 10 moves the decimal one place to the left,
dividing by 100 moves it two places to the left and so on.  We can
think of dividing by 10 as multiplying by 1/10, and the way to write
1/10 using exponents is 10^(-1).  So multiplying with a negative
exponent on the 10 tells us to move the decimal to the _left_ that
many places, and makes the expanded number smaller:

  4.56 x 10^(-3) = 0.00456 since the decimal moves 3 places to the
  left

As you can see, writing very large or very small numbers is much
easier with scientific notation:

  456,000,000,000,000,000,000,000 is just 4.56 x 10^22
  0.00000000000000000000000000456 is just 4.56 x 10^(-27)

Count the places I moved left or right on those two to confirm the
exponents I used.

So, how do we multiply two numbers that are written in scientific
notation?  Let's keep in mind that multiplication is an associative
operation--in other words, when several things are being multiplied
together it doesn't matter what order you multiply them in:

  2*(3*4) = 2*(4*3) = (3*4)*2 = 4*(2*3) = 24

When multiplying two numbers in scientific notation, there are really
four quantities being multiplied.  For instance:

  (3.2 x 10^2)(4.1 x 10^8) = (3.2)*(10^2)*(4.1)*(10^8)

Since we can multiply those four things in any order, we do it this way:

  [(3.2)*(4.1)] * [(10^2)*(10^8)]

Multiply 3.2 and 4.1 to get 13.12, and then multiply the two powers of
10.  Remember that when multiplying two exponent expressions with the
same base, we keep the base and add the exponents since:

  10^2 * 10^4 = 10*10 * 10*10*10*10 = 10*10*10*10*10*10 = 10^6

So, putting all this together, we now have:

  [(3.2)*(4.1)] * [(10^2)*(10^8)] = 13.12 x 10^10

But we're not quite done.  Remember that our answer needs to be in
scientific notation, too, and that means only one digit before the
decimal.  So if we would move 10 places to the right from 13.12, we
would need to move 11 places to the right from 1.312.  Our final
answer is:

  (3.2 x 10^2)(4.1 x 10^8) = 13.12 x 10^10 = 1.312 x 10^11

I've tried to explain it all in detail rather than just showing you
how because it's important to understand _why_ we do it the way we do.
Basically, the idea is to multiply that two numbers that are in front,
add the powers of 10, and adjust the answer to put it in scientific
notation.  Here are a few more examples:

  (8 x 10^9)*(2.1 x 10^12) = 16.8 x 10^21 = 1.68 x 10^22

  (5.2 x 10^[-9])*(4.5 x 10^[-6]) = 23.4 x 10^[-15] = 2.34 x 10^[-14]

  (6.8 x 10^[-7])*(3.2 x 10^10) = 21.76 x 10^3 = 2.176 x 10^4

Just keep in mind how the decimal needs to move when you are adjusting
your final answer.  If you were multiplying three or more numbers in
scientific notation, the process is the same.

Hope this helps.  Feel free to write back if you have more questions
or if anything I said is unclear.

- Doctor Riz, The Math Forum
  http://mathforum.org/dr.math/