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/
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