Question

Evaluate and write your answer to scientific notaion: (7 x 10^-2)(15 x 10^3).

My answer: I got (1.05 x 10^-4) , can someone verify my answer because I'm not to sure about this one

Answer 1

please help!

Answer 2

You want us to verify that your answer is correct, or that it's your answer? :-)

Unfortunately, it is not correct. Here's what you do:

each number in scientific notation has two parts, the mantissa, or fractional part, and the exponent. For your first one, the mantissa is 7 and the exponent is -2. Similarly, for the second number, the mantissa is 15 and the exponent is 3.

To multiply two numbers in scientific notation, first multiply the mantissas. You appear to have done this correctly. Next, combine the exponents by adding. Here you went astray, it seems. -2 + 3 = 1. Remember, the \(10^{-2}\) represents 0.01 and the \(10^3\) represents 1000. Your problem could instead be written as
\[(7*0.01)* (15*1000)\]Because multiplication is commutative (order doesn't matter), we could rearrange that as \[(7*15)*(0.01*1000) = (105)*(10)\]which is how I proposed that you do the computation. We're almost done! Now we have to normalize the mantissa, which is to say we want to multiply or divide by 10 repeatedly until the mantissa is between 1 and 10 if positive, or -1 and -10 if negative. \(1 \le | m | \lt 10\), if \(m\) is the mantissa. Our mantissa is 105, so it needs adjusting. Every time we can factor out a 10 from it (divide by 10), we increase the exponent by 1 to keep the value of the whole thing constant. Here's how it looks:
\[105*10^1 = 10.5*10^2 = 1.05 * 10^3\]
Now our answer is in the proper normalized form. In some circles, normalization is not required; I can't say whether your instructor expects it or not. However, if you know how to do it, you'll be prepared for the possibility!

If our mantissa turns out to be < 1, we multiply by 10 in the same fashion, this time decreasing the exponent by 1 for each factor of 10 we put in. For example,
\[\frac{(2*10^1)}{(4 * 10^3 )} = 2/4 * 10^{-2} = 0.5 * 10^{-2} = 5*10^{-3}\]

Answer 3

Thanks

Answer 4

Sometimes it's more convenient not to normalize the answer. For example, in science and engineering applications, it may be more convenient to have the number have an exponent which is a multiple of 3, like 10^6 or 10^9. Much easier to read off the appropriate SI prefix for the number that way: compare 15,264 meters written in scientific notation: 1.5264 * 10^4 meters or 15.264 * 10^3 meters. The last one is easily seen to be 15.264 kilometers, the previous one is not so clear. Using an exponent that is a multiple of 3 is called engineering notation, and some calculators have a mode that will present all of your answers that way.