Hi L,
Let's break this problem down.
1. (4.2*10^{2})(4.2*10^{5})
First let's write out the whole problem:
(4.2*10*10)(4.2*10*10*10*10*10)
which is the same as 4.2*10*10*4.2*10*10*10*10*10.
You can mix-up the order of the numbers here because all you are doing is multiplying and therefore it won't change the result of the problem, and it's better to do that here to put the numbers in an order so you can see and understand them more clearly. So next order the numbers like this:
4.2*4.2*10*10*10*10*10*10*10.
A verbal explanation of this: You can see here what the original problem is doing is multiplying a string of numbers, that is it's multiplying 4.2 by itself once, and you are multiplying that by 10 5 times. (You are not multiplying 10 by 5 but 10 by itself 5 times to the first 4.2 * 4.2. Follow?)
So you are squaring 4.2 and then multiplying that by 10 to the fifth power (10^{5}) because you are multiplying 10 by itself 5 times.
4.2 * 4.2 then is 17.64.
And 10^5
Well
10^1 = 10
10^2 = 100
10^3 = 1,000
10^4 = 10,000 and
10^5 = 100,000
so you are multiplying 17.64 by 100,000 which is 176,400,000 (one hundred seventy-six million, four hundred thousand).
2. Starting with this problem, you need to know that people use scientific notation to show really big or really small numbers in a simple way.
It's also good to know about exponential increase and exponential decrease for the next two problems.
A number in scientific notation is a number between 1 and 10 followed by a decimal point and some numbers that follow that decimal point, like 4.234567 or 5.24345. This number with a decimal after it is then multiplied by a number raised to a positive power (which will increase that number exponentially, like you can see in the problem 10^5 above) or a negative number, (which decreases that number exponentially. Exponential decrease doesn't mean that it will become negative, but that it will get closer and closer to 0 because it will be smaller than 1 so for example 10^-1 = 1/10, 10^-2 = 100, 10^-3 = 1/1,000, etc.).
In the second problem, you are raising the whole expression (2.5 * 10^{-2}) to the fourth power, so before you do that, you need to figure out what the expression in the parentheses means.
Well you are multiplying 2.5 by 10^-2, and what does it mean when you raise something to a negative power?
It means that you raise the inverse of that number to that power, so here you will invert 10 (to invert means to flip) (remember a number is the same thing as that number over 1, so 10 = 10/1), so when you invert it, it becomes 1/10. Afterwords you can raise it to the second power, or square it.
(1/10)^2 = (1/10)*(1/10)
So you multiply the numerators with the numerators, and the denominators with the denominators (when you multiply, it's real easy, you multiply straight across),
you multiply 1 by 1 for the numerator, and 10 by 10 for the denominator, and you get 1/100.
so 2.5 * 1/100 is the same as
2.5/1 * 1/100, and you multiply straight across
so you multiply 2.5 by 1 (for the numerator) and 1 by 100 for the denominator and you get
2.5/100 which is a meager
.025 (or 2.5%)
Next you have to raise this to the fourth power.
When you multiply a decimal by a decimal, what you are trying to find out then is what a part of a part is, so the number will become smaller...and smaller and smaller because you are multiplying .025 by a decimal (that is, by itself) 3 more times (for a total of 4 times because .025 = .025 * 1).
.025 * .025 * .025 *.025 (or .025^4) is 0.000000390625, and to write that in scientific notation you move the decimal point 7 places to the right (so you are multiplying that number by 1,000,000), and you get 3.90625. You need to show that this number is really the original 0.000000390625 though, so you multiply it by 10^{-7} and you get the original number .000000390625. (Remember 10^{-7} is 10 flipped to the 7th, in other words, it stands for one millionth, or one out of a million to show that your answer 3.90625 is really one millionth of what you wrote it as.)
So the answer for this problem is 3.90625 * 10^-7.
So you can probably guess how you will solve the last problem.
5,100,000 - You need to get it to be 5.1, and to get it to be 5.1, you need to show that it is 100,000 times (one hundred thousand times) greater than 5.1, so you do that by multiplying 5.1 by 10^{5}.
I hope this is good.