Question

Write as a number in standard notation: 9 x 10^7
A. 90,000,000
B. 9,000,000
C. 900,000
D. 90,000

Answer 1

A

Answer 2

Scientific notation is, essentially, a method for writing really big or really small numbers. It is called scientific notation because these huge numbers are often found in scientific work, like the size of an atom or The absolute value of a number is its distance from 0 on a number line. For example, the number 9 is 9 units away from 0. Therefore its absolute value is 9.

Negative numbers are more interesting, because the number -4 is still 4 units away from 0. The absolute value of -4 is therefore positive 4.

The absolute value leaves a positive unchanged, and makes a negative positive.

An absolute value is written like this: |x|, and is read as "the absolute value of x." Note: In certain places, such as calculator and computer programs, you may see it written as abs(x), which naturally means "the absolute value of x," but |x| is the accepted way to write it on your homework and tests.

To force a number to be negative, you can write -|x|. This takes the number, makes it positive, and then negates it. Remember -- just putting a negative sign in front of a number doesn't make it negative. If the number was already negative then you just made it positive! Using the absolute value guarantees we have a positive value so that the negative sign will definitely make it negative.

Examples:

|4| = 4
|-4| = 4
|4+3| = 7
|-4-3| = 7
|3-4| = 1
-|4| = -4
-|-4| = -4



The absolute value sign can be used in equations as well:
|-8| = x, thus x=8
|x| = 8, thus x=8 or x=-8. Remember that |-8| is also 8 so there are two solutions here!
|x| = -8, there are no solutions because the absolute value can never be negative.

Absolute values are easy enough to compute when they contain constants (regular numbers), but absolute value equations containing variables are more difficult. Suppose we are given the following equation, and asked to solve for x:
|x+2| = 9

We can not assume that x+2 is positive or negative, so we can not simply "drop the bars." If x+2 were indeed negative, the absolute value of x+2 would really be -(x+2), since a negative times a negative equals a positive. We will solve using cases.

The first case, or possibility, is that x+2 is positive. Taking the absolute value of a positive does not change the outcome.
First Case: x + 2 = 9

The second case is that x+2 is negative. To get the absolute value of a negative, you have to negate it (which makes it positive again). Therefore |x+2| = -(x+2).
Second Case: -(x + 2) = 9

Here we can solve both cases for x.
x+2 = 9
x = 7

or

-(x+2) = 9
-x -2 = 9
-x = 11
x = -11

Our two solutions for |x+2|=9 are 7 and -11. Try them. They both work.

More complicated equations can usually be solved the same way, by splitting the absolute value into two cases. You should check that you answers match the case, however. If you get a possible answer of 8 from the negative case, that can't be right.

If you have time you should plug your answers back into the original equation to check for correctness.

That's about it for a simple introduction of the absolute value. You may not use it that often, but it is vital to understand later in math. For more information, try searching Google for "absolute value." You may also want to check out another lesson on absolute value provided by PurpleMath, or perhaps this one from Wikipedia.

(c) 2007 Ted Wilcox

the mass of the earth.

For example, you might have the number 6000000000000. That’s really big, right? Unfortunately, it isn’t easy to tell exactly how big at first glance with all those zeroes stuck on the end. Instead, the number could be written as 6 * 1000000000000. Then you can change the 1000000000000 to an easier to understand number: 10^12. Putting it all together, we have 6 * 10^12. Now you can compare that number to others, because the 10^12 means there are 12 zeroes at the end.

The value of scientific notation becomes clear when you try to multiply or divide these numbers. What is 50000000 * 3000000? You could do this relatively easily by multiplying 3 times 5 and then adding up all the zeroes, but that still takes time, and you could easily miscount all the zeroes. Instead, scientific notation allows us to multiply 5 * 10^7 times 3 *10^6. You multiply the 5 and the 3 to get 15, and then add the exponents on the 10’s. The answer is 15*10^13. However, in order for scientific notation to be completely correct, the number at the beginning must be between 1 and 10. The 15 has to be changed into 1.5, and to make up for this we multiply the whole thing by another factor of 10, giving 1.5*10^14.

Scientific notation can also be used for very small numbers in much the same way. 0.000005 is written as 5*10^-6, because you use negative exponents on the 10 when the number is very small. Remember, negative exponents do not make the number negative, but just very small. Try multiplying .00009 * .00003. The numbers in scientific notation are 9*10^-5 times 3*10^-5. The answer is computed the same way as before, yielding 27*10^-10, or 2.7*10^-9. Here are a few more examples to illustrate the principles of scientific notation:

5*10^3 = 5000
8*10^-1 = .8
7*10^-3 = .007

(5*10^8)(4*10^15) = 2 * 10^24
(5*10^-8)(4*10^15) = 2 * 10^8

(9*10^6) / (3*10^4) = 3*10^2 (when dividing, you subtract the exponents)
(8*10^-5) / (2*10^-3) = 4*10^-2The absolute value of a number is its distance from 0 on a number line. For example, the number 9 is 9 units away from 0. Therefore its absolute value is 9.

Negative numbers are more interesting, because the number -4 is still 4 units away from 0. The absolute value of -4 is therefore positive 4.

The absolute value leaves a positive unchanged, and makes a negative positive.

An absolute value is written like this: |x|, and is read as "the absolute value of x." Note: In certain places, such as calculator and computer programs, you may see it written as abs(x), which naturally means "the absolute value of x," but |x| is the accepted way to write it on your homework and tests.

To force a number to be negative, you can write -|x|. This takes the number, makes it positive, and then negates it. Remember -- just putting a negative sign in front of a number doesn't make it negative. If the number was already negative then you just made it positive! Using the absolute value guarantees we have a positive value so that the negative sign will definitely make it negative.

Examples:

|4| = 4
|-4| = 4
|4+3| = 7
|-4-3| = 7
|3-4| = 1
-|4| = -4
-|-4| = -4



The absolute value sign can be used in equations as well:
|-8| = x, thus x=8
|x| = 8, thus x=8 or x=-8. Remember that |-8| is also 8 so there are two solutions here!
|x| = -8, there are no solutions because the absolute value can never be negative.

Absolute values are easy enough to compute when they contain constants (regular numbers), but absolute value equations containing variables are more difficult. Suppose we are given the following equation, and asked to solve for x:
|x+2| = 9

We can not assume that x+2 is positive or negative, so we can not simply "drop the bars." If x+2 were indeed negative, the absolute value of x+2 would really be -(x+2), since a negative times a negative equals a positive. We will solve using cases.

The first case, or possibility, is that x+2 is positive. Taking the absolute value of a positive does not change the outcome.
First Case: x + 2 = 9

The second case is that x+2 is negative. To get the absolute value of a negative, you have to negate it (which makes it positive again). Therefore |x+2| = -(x+2).
Second Case: -(x + 2) = 9

Here we can solve both cases for x.
x+2 = 9
x = 7

or

-(x+2) = 9
-x -2 = 9
-x = 11
x = -11

Our two solutions for |x+2|=9 are 7 and -11. Try them. They both work.

More complicated equations can usually be solved the same way, by splitting the absolute value into two cases. You should check that you answers match the case, however. If you get a possible answer of 8 from the negative case, that can't be right.

If you have time you should plug your answers back into the original equation to check for correctness.

That's about it for a simple introduction of the absolute value. You may not use it that often, but it is vital to understand later in math. For more information, try searching Google for "absolute value." You may also want to check out another lesson on absolute value provided by PurpleMath, or perhaps this one from Wikipedia.

(c) 2007 Ted Wilcox
Numbers that can be written as fractions are called rational numbers. Here's a small list: 1/2, 4/7 and 2/9 are samples of rational numbers. The numbers 0.5, .571428......., .2222.... are all rational numbers because they are exactly the same as the fractions just listed ( and the definition of rational means they can be rewritten as fractions ).

What is a rational function? A rational function is a polynomial function divided by another polynomial function. Here is a small list of what they look like:

1) f(x) = x^3 - 2x +1/3/3x - 1

2) y = x^2 - 3x - 2/3x - 2

What does it mean to express a function in standard form? To write a function in standard form, simply replace the letter y with the function notation requested.

Sample: Express y = x^2 - 3x - 2/3x - 2 in standard form.

Since y = f(x) and f(x) = y, we can replace y with f(x).

Answer: f(x) = x^2 - 3x -2/3x - 2.

The general form for rational functions looks like this: f(x) = h(x)/g(x). This general form simply means that a rational function equals the ratio one polynomial function divided by another polynomial function.

How to recognize a rational function?

HINT: If you see a fraction or fractions that include a variable or variables in the denominator, then you are looking at a rational function.

Sample: 90/m + 30/n = 20 is a rational function because the variables m and n appear in the denominator of both fractions.

Also, learn how to multiply and divide rational functions.

By Mr. Feliz
(c) 2005Numbers that can be written as fractions are called rational numbers. Here's a small list: 1/2, 4/7 and 2/9 are samples of rational numbers. The numbers 0.5, .571428......., .2222.... are all rational numbers because they are exactly the same as the fractions just listed ( and the definition of rational means they can be rewritten as fractions ).

What is a rational function? A rational function is a polynomial function divided by another polynomial function. Here is a small list of what they look like:

1) f(x) = x^3 - 2x +1/3/3x - 1

2) y = x^2 - 3x - 2/3x - 2

What does it mean to express a function in standard form? To write a function in standard form, simply replace the letter y with the function notation requested.

Sample: Express y = x^2 - 3x - 2/3x - 2 in standard form.

Since y = f(x) and f(x) = y, we can replace y with f(x).

Answer: f(x) = x^2 - 3x -2/3x - 2.

The general form for rational functions looks like this: f(x) = h(x)/g(x). This general form simply means that a rational function equals the ratio one polynomial function divided by another polynomial function.

How to recognize a rational function?

HINT: If you see a fraction or fractions that include a variable or variables in the denominator, then you are looking at a rational function.

Sample: 90/m + 30/n = 20 is a rational function because the variables m and n appear in the denominator of both fractions.

Also, learn how to multiply and divide rational functions.

By Mr. Feliz
(c) 2005If 3 players are selected from a team of 9, how many different combinations are possible? First, we need to define what a combination means. In mathematical terms, a combination is an subset of items from a larger set such that the order of the items does not matter. For example, if John, Fred, and Bill are selected from the team, that is considered the same combination as Fred, John, and Bill.

Now, how do we calculate the number of possible combinations? Use the following formula:



First, let me explain the notation on the left. That means that from a group of n objects, we are selecting r of them. It's just a standard notation used for combinations, but you might also see something like nCr used instead.

The rest of the formula is more straightforward. Remember that the ! means factorial, the product of that number and all positive integers less than it. For example, 4! = 4*3*2*1.

In the original example I gave, we wanted the number of combinations when selecting 3 out of 9:



Notice how a good portion of the multiplication cancelled out. There's no need to calculation 9! all the way out when the 6! below it will just cancel out everything except 9*8*7. In the end, we see that there are 84 ways to pick 3 people from a group of 9 as long as order does not matter.

Consider another example. If a traveller has the option of visiting any 6 of the 50 United States, but doesn't care in which order he sees them, how many different trips are possible? Ignore, of course, the fact that many of these combinations would be very random and unnecessarily complicated!

The problem is a simple combination. There are 50 total states, and we must select 6:



There are an astonishing 15 million different groups of 6 states!

Hopefully this gets you started with combinations. For more help, consider these other lessons on combinations, or this page that explains the difference between a combination and a permutation.

So yes it should be (A) so heres any note's for u !!!!!!!!!