3^-3x10^-3=Answer 30^-3 does not work
Answers
Step-by-step explanation:
Basic Math: Scientific Notation
In this section, you will occasionally be asked to answer some questions. Whenever a problem set is given, you should answer the questions on a separate sheet of paper and then verify your answers by clicking on "Answers."
The first thing to learn is how to convert numbers back and forth between scientific notation and ordinary decimal notation. The expression "10n", where n is a whole number, simply means "10 raised to the nth power," or in other words, a number gotten by using 10 as a factor n times:
105 = 10 x 10 x 10 x 10 x 10 = 100,000 (5 zeros)
108 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000 (8 zeros)
Notice that the number of zeros in the ordinary decimal expression is exactly equal to the power to which 10 is raised.
If the number is expressed in words, first write it down as an ordinary decimal number and then convert. Thus, "ten million" becomes 10,000,000. There are seven zeros, so in powers of ten notation ten million is written 107.
A number which is some power of 1/10 can also be expressed easily in scientific notation. By definition,
1/10 = 10-1 ("ten to the minus one power")
More generally, the expression "10-n" (where n is a whole number) means ( 1/10 )n. Thus
10-3 = ( 1 / 10 )3 = 1 / ( 10 x 10 x 10) = 1/1000
10-8 = ( 1 / 10 )8 = 1/100,000,000
Scientific notation was invented to help scientists (and science students!)deal with very large and very small numbers, without getting lost in all the zeros. Now answer the following on a separate sheet of paper and check your answers by clicking on "Answers":
First Problem Set
Express 1-6 in scientific notation, and 7-10 in ordinary notation:
1. 100
2. 10,000,000
3. 1 / 10,000
4. one million
5. 1 / 10,000,000
6. one ten millionth
7. 103
8. 10-5
9. 109
10. 1 X 10-2
Answers
What about numbers that are not exact powers on ten, such as 2000, 0.0003, etc.? Actually, they are only a little more complicated to write down than powers of ten. Take 2000 as an example:
2000 = 2 x 1000 = 2 x 103
As another example, take 0.00003, or "three ten-thousandths":
0.0003 = 3 x 1 / 10,000 = 3 x 10-4
There is a simple procedure for getting a decimal number into the "standard form" for scientific notation:
First, write down the number as the number itself times 100. This can be done because 100 equals one, and any number times one equals that number. The number is now in the standard form:
coefficient x 10 exponent
Second, start moving the decimal point in the coefficient to the right or left. For each place you move the decimal place to the left, add 1 to the exponent. For each place you move it to the right, subtract 1 from the exponent. What you are doing is dividing (or multiplying) the coefficient by 10 each time, while at the same time multiplying (or dividing) the exponent term by 10 each time. Since what you do to the exponent term undoes what you do to the coefficient, the total number does not change.
Some examples will hopefully make it clear:
2000 = 2000 x 100= 200 x 101= 20 x 102= 2 x 103
0.0003 = 0.0003 x 100= 0.003 x 10-1 = 0.03 x 10-2 = 0.3 x 10-3= 3 x 10-4
You should move the decimal point until there is exactly one nonzero digit to the left of the decimal point, as in the last case of each example given. We then say that the number is fully in the standard form. You should always express scientific notation numbers in the standard form. Notice that you don't really have to write down each of the steps above; it is enough to count the number of places to move the decimal point and use that number to add or subtract from the exponent. Some examples:
250,000 = 2.5 x 105 5 places to the left
0.000035 = 3.5 x 10-5 5 places to the right
0.00000001 = 1 x 10-8 = 10-8 8 places to the right
Step-by-step explanation:
We know there are 14, but how do we write this calculation? If we just write
2 + 3 x 4
how does a reader know whether the answer is
2 + 3 = 5, then multiply by 4 to get 20 or
3 x 4 = 12, then 2 + 12 to get 14?
There are two steps needed to find the answer; addition and multiplication. Without an agreed upon order of when we perform each of these operations to calculate a written expression, we could get two different answers. If we want to all get the same "correct" answer when we only have the written expression to guide us, it is important that we all interpret the expression the same way.
One way of explaining the order is to use brackets. This always works. To say that the 3 x 4 is done before the adding, we would use brackets like this:
2 + (3 x 4)
The brackets show us that 3 x 4 needs to be worked out first and then added to 2. However, we can also agree on an order of operations, which is explained below.
Another example: Calculate 15- 10 ÷ 5
If you do the subtraction first, you will get 1. If you do the division first, which is actually correct according to the rules explained below, you will get 13. We need an agreed order.
division first (correct) subtraction first blue indicates the operations being worked on first
15 - 10 ÷ 5 15 - 10 ÷ 5
= 15 - 10 ÷ 5 = 15 - 10 ÷ 5
= 15 - 2 = 13 = 5 ÷ 5 = 1
How to use brackets
Brackets are marks of inclusion which tell us which parts of an expression go together. We use brackets in an expression to indicate which part to calculate first. It can be useful to think of brackets as a circle with the top and bottom deleted to remind you that brackets indicate that everything inside the 'circle' is self-contained and must be worked out first. Although brackets usually look like ( ), brackets can also look like { } or [ ] and need to be treated in the same way. Brackets are sometimes referred to as "parentheses".
want division first want subtraction first blue indicates the operations being worked on first
15 - (10 ÷ 5) (15 - 10) ÷ 5
= 15 - (10 ÷ 5) = (15 - 10) ÷ 5
= 15 - 2 = 13 = 5 ÷ 5 = 1
There are more examples on how to use brackets in complicated examples below.
If we used brackets consistently we would not have to be concerned with the order of operations. We could just work from innermost brackets outwards to eventually get our answer. However using lots of brackets can become tedious and confusing, as in the following example, so we need some agreed rules.
3 + ((4÷2)x7)-(6÷3)-((4x2)+((8÷2)+(3x3)))
You can check how to work out this monster by clicking here, but the next section tells you how to avoid the worst monsters.