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evaluate expression

how do I evaluate the expressions, then write the answer in scientific notation?
 
1.  (4.2 x 10^2) (4.2 x 10^5)
 
2.  (2.5 x 10 ^-2)^4
 
 
 
write this number only in scientific notation
 
1. 5,100,000
 
 
Thank you
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8 Answers

1. (4.2 x 10^2) (4.2 x 10^5)
 
Rewrite as
 
(4.2 x 4.2)[10(2+5)] = 17.64 x 107 or, more properly, 1.764 x 108
 
2. (2.5 x 10 ^-2)^4
 
Rewrite as
 
2.5 x 10^[(-2)(4)] = 2.5 x 10(-8) = 7.90625 x 107
 
1. 5,100,000
 
Count spaces to the right of 5.  There are six.  So 5,100,000 = 5.1 x 106
1)(4.2*10^2)(4.2*10^5)=(4.2*4.2)(10^2*10^5)
                                      =(17.64)(10^7)
                                      =176,400,000
or
420*420,000=176,400,000
176,400,000=1.764*10^8
 
2)(2.5*10^-2)^4=(2.5^4)(10^-8)
                          =39.0625*10^-8
                          =0.000000390625
 
1)5,100,000=5.1*10^6
1. (4.2)(4.2) x 10 ^ (2+3) = 17.64 x 10^6 = 1.764 x 10^7 ~ 1.8 x 10^7
2. (2.5 x 10^-2)^4 = (2.5^4) x (10^-2)(^4) = 33.1776 x 10 ^ -8 = 3.31776 x 10^-7 ~ 3.3 x 10^-7
 
3. 5,100,000 = 5.1 x 10^6
1.  To evaluate (4.2 x 10^2) (4.2 x 10^5), you multiply the mantissa (or digits) of each number: 4.2 x 4.2 = 17.64, and add the exponents of 10: 2+5 =7, resulting in 17.64 x 10^7.  Simplify so that the mantissa is less than 10, which means you also add 1 to the exponent, giving 1.764 x 10^8.  
 
2.  For  (2.5 x 10 ^-2)^4, first you evaluate 2.5^4 = 39.0625.  When you raise an exponent to another power, you multiply the exponent by the powere: (10^-2)^4 = 10^-8.  So you have 39.0625 x 10-8.  Simplifying so that the mantissa is less than 10 gives you 3.90625 x 10^-7.  (39 = 3.9x10^1)
 
1. 5,100,000 = 5.1 x 10^6.  The mantissa (5.1) should be greater than 1, but less than 10.  If you're moving the decimal point to the left, the exponent of 10 is positive.  
When dealing with exponents, you need to remember PEMDAS and step-down. When you multiply factors with exponents, you "step down" when dealing with the exponent and would add them. When you raise a power to another power, you "step down" and multiply exponents.
 
So, (4.2 x 10^2)(4.2 x 10^5) would be the same as (4.2 x 4.2)(10^2 x 10^5) since multiplication is associative. This would be (17.64)(10^7).
 
Now scientific notation requires the coefficient to be at least 1 and less than 10 (only one digit allowed to the left of the decimal), so 17.64 = 1.764 x 10^1
 
Now you have (1.764 x 10^1)(10^7). Again associative property gives you (1.764)(10^1 x 10^7)
= 1.764 x 10^8
 
Now if you are worried about significant figures, you would be limited to 2 digits,
so it would now be 1.8 x 10^8
 
Now with (2.5 x 10^-2)^4. Distribution gives you (2.5)^4 x (10^-2)^4
 
39.0625 x (10^-2)^4
 
With the "step down" (10^-2)^4 becomes 10^-8 (multiply powers raised to powers).
 
39.0625 x 10^-8 =3.90625 x 10^-7 (see previous explanation for coefficient which would be +1-8). With sig figs, you have 3.9 x 10^-7
 
5.1 x 10^6 (you count the number of places the decimal is moved, left is positive and right s negative).
 
Hope this helps
 
Mike
 4.2 * 4.2 * 10^2 * 10^5 =
 
  ( 4.2 ^ 2) * 10 ^7 =
    17.64 * 10 ^7
 
 2,
       (2.5 * 10 ^-2) ^4 =
 
         2.5 ^ 4 * 10 ^-8 =
     
         39.06 25  = 390625  = 39. 0625
          10 ^ 8        10 ^4
 
  3.     1.5 10000 = 1.51
         
         Probably it was   15, 100, 000 = 1.5* 10^7
 
   
Hi L,
 
Let's break this problem down.
 
1. (4.2*102)(4.2*105)
 
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 (105) 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 105.
 
I hope this is good.
Scientific notation basically puts large numbers into the form a e b where a and b are numbers. For example:
1000 = 1e3 (three zeroes)
You can also have decimals:
2200 = 2.2e3. Think of it as moving the decimal place three to the right if positive, to the left if negative.
 
So,  let's convert the numbers: (I'm using * so not to confuse x with variable x's.)
4.2 * 10^2 = 4.2e2
4.2 * 10^5 = 4.2e5
then 4.2e2 * 4.2e5 = 17.64e7 (moving decimal over two and then five).
 
Likewise (2.5 * 10^-2)^4 = 2.5^4 * (10^-2)^4 = 39.0625e-8 (decimal place moves two to the left, four times).
 
Since scientific notation only allows one significant digit before the decimal, we'll have to do some manipulation. It looks kind of like this: 39*1e-8=39*(10^-1)*(10^1)e-8=3.9e-7. Tricky eh?
 
To write a 5,100,000 in scientific notation, move the decimal until there's only one significant digit before it. that would move the decimal point six times to make 5.1 e6.