1 Million Divided By 10000

In this section, you will occasionally be asked to respond some questions. Whenever a problem gear up is given, you should reply the questions on a separate sheet of newspaper and then verify your answers past clicking on "Answers."

The first thing to acquire is how to convert numbers dorsum and along between scientific notation and ordinary decimal notation. The expression "10ⁿ", where n is a whole number, just ways "10 raised to the nth power," or in other words, a number gotten by using 10 equally a factor n times:

10⁵ = ten x x ten 10 x 10 x 10 = 100,000 (v zeros)

ten⁸ = x 10 x x 10 x x x ten 10 10 x 10 x 10 = 100,000,000 (8 zeros)

Discover that the number of zeros in the ordinary decimal expression is exactly equal to the power to which x is raised.

If the number is expressed in words, beginning write it downwardly as an ordinary decimal number so convert. Thus, "ten million" becomes 10,000,000. There are seven zeros, then in powers of ten notation ten million is written 10⁷.

A number which is some power of 1/ten can also be expressed easily in scientific notation. Past definition,

ane/x = x^-1 ("ten to the minus one ability")

More generally, the expression "x^-n" (where n is a whole number) means ( 1/10 )ⁿ. Thus

10^-iii = ( ane / 10 )ⁱⁱⁱ = 1 / ( 10 x 10 x 10) = 1/1000

10^-viii = ( 1 / 10 )⁸ = ane/100,000,000

Scientific notation was invented to help scientists (and scientific discipline students!)deal with very large and very small numbers, without getting lost in all the zeros. At present answer the following on a carve up sheet of paper and check your answers by clicking on "Answers":

Beginning Problem Ready

Express 1-half-dozen in scientific note, and 7-10 in ordinary notation:

i. 100	ii. ten,000,000
3. 1 / 10,000	four. i million
5. 1 / x,000,000	half dozen. one ten millionth
7. x^three	8. ten^-5
nine. ten⁹	10. 1 Ten 10^-2

Answers

What well-nigh numbers that are non exact powers on x, such every bit 2000, 0.0003, etc.? Actually, they are only a footling more than complicated to write down than powers of ten. Take 2000 as an case:

2000 = 2 x yard = 2 x 10³

Equally another example, take 0.00003, or "three 10-thousandths":

0.0003 = 3 x 1 / 10,000 = three x 10^-4

At that place is a simple process for getting a decimal number into the "standard form" for scientific notation:

First, write down the number every bit the number itself times 10⁰. This can exist done because 10⁰ equals one, and any number times i equals that number. The number is now in the standard class:

coefficient x 10 ^exponent

Second, start moving the decimal signal in the coefficient to the right or left. For each identify y'all move the decimal place to the left, add 1 to the exponent. For each place you move it to the correct, subtract 1 from the exponent. What you are doing is dividing (or multiplying) the coefficient past 10 each time, while at the aforementioned fourth dimension multiplying (or dividing) the exponent term by ten each time. Since what you lot 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 10 10⁰= 200 ten 10^one= 20 x 10²= 2 ten 10³

0.0003 = 0.0003 x ten⁰= 0.003 x 10^-i = 0.03 ten 10^-2 = 0.three x 10^-3= 3 ten ten^-four

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 case given. We and so say that the number is fully in the standard form. You should always limited scientific annotation numbers in the standard form. Detect that you lot don't really have to write down each of the steps above; it is enough to count the number of places to motion the decimal betoken and use that number to add or decrease from the exponent. Some examples:

250,000 = 2.5 ten x^v 5 places to the left

0.000035 = iii.five ten 10^-v 5 places to the correct

0.00000001 = 1 10 10^-8 = 10^-viii viii places to the correct

Second Problem Gear up

Express 1-6 in scientific notation, and 7-10 in ordinry annotation:

1. 342,000,000	2. 0.000923
3. eight million	4. 0.0000064
five. 47,682	six. 0.0249
7. iv x 10^vii	eight. iii.22 x 10^-3
9. 8.4 x ten^ten	ten. 6.33 x 10^-half-dozen

Answers

The most difficult kind of calculation that tin be done with numbers expressed in scientific notation turns out to be improver or subtraction. Multiplication, segmentation, and raising to powers is actually easier. Then, we'll deal with these showtime.

The rule for multiplying two numbers expressed in scientific annotation has three steps:

Multiply the coefficients to get the new coefficient.

Add together the exponents (scout the signs!) to get the new exponent.

Get the number into the standard form, if needed.

Examples:

(4 10 10ⁱⁱⁱ) x (2 x 10⁷) = ( 4 x 2 ) 10 ( 10^{3 + seven} ) = 8 x 10¹⁰

(two x 10^-five) ten (2.5 ten 10^eight) = ( two 10 2.5 ) ten ( x^{-5+ 8} ) = 5 ten 10^three

(three x ten^-seven) x (3 x 10^-8) = ( 3 x 3 ) ten ( 10 ^{-7 + (-8)} ) = 9 x 10^-xv

(4 x 10^seven) x (iii ten 10⁵) = ( 4 ten 3 ) ten ( 10 ^{7 + v} ) = 12 x 10¹²= 1.two x 10¹³

The steps for division are similar:

Divide the coefficients to go the new coefficient

Decrease the "bottom" exponent from the "top" i (really watch the signs!) to get the new exponent.

Get the number into the standard form, if needed.

Some examples:

(six 10 10⁵) / (2 ten 10³) = ( 6 / ii ) 10 ( ten ^{5 - 3} ) = 3 x x²

(9 x x⁸) / (3 x 10^-v) = ( ix / iii ) 10 ( 10 ^{eight - (-five)} ) = 3 ten ( 10 ^{eight + 5} ) = three x 10^thirteen

(5 x x³) / (2 x x⁷) = ( v / 2 ) 10 ( ten ^{iii - 7} ) = 2.5 x 10^-4

(ii 10 10⁵) / (4 x 10²) = ( ii / 4 ) x ( 10 ^{v - 2} ) = 0.5 x ten³= 5 ten 10²

If you are given a number in scientific notation to raise to a power, remember that all this means is that it is used equally a cistron that many times. Only write the number downwards equally many times every bit the power to which it is to exist raised, and utilise the rules for multiplication repeatedly.

Example:

(2 x 10⁵)³ = (2 10 10⁵) x (2 x 10^five) x (2 x ten⁵)

(ii x 10⁵)^three = ( ii x 2 x 2 ) 10 ( 10^v x x⁵ 10 x^v)

(2 x 10⁵)^three = 8 x ( 10 ^{five + 5 + five} ) = 8 ten 10¹⁵

In a situation where you have to raise things to a ability and do multiplication or division, always stop raising to the power first, then practice the other operation.50

Case:

(ii x 10^ix)³ / (6 ten 10^-2)²

(2 ten 2 x 2) x (10⁹ 10 10^ix x ten⁹) / (vi x 6) x (ten^-2 ten 10^-2)

(eight x 10 ^{nine + ix +nine}) / (36 10 ten ^{-2 -2})

(8 ten ten²⁷)/ (36 x 10^-4)

(8/36) ten (10^{27 - (-4)})

0.22 10 10 ³¹ = ii.ii x 10 ³⁰

Third Problem Set

Calculate the following:

1. (seven 10 x⁶) / (2 x 10⁴)	2. (ii x x^vii) x (4 10 ten^-9)
3. (5 x 10⁸) 10 (5 x tenⁱⁱⁱ)	4. (vi x 10ⁱⁱⁱ)³ / (3 10 10⁶)^four
5. (v x 10ⁱⁱⁱ) / (2 x ten³)	six. (3 x 10⁴)²
seven. (4 x 10^-vi)³	8. (2 x 10⁵) x (half-dozen x 10⁷) / (four x x⁸)

Answers

Add-on and subtraction are a little more involved. In that location are four basic steps:

Discover the number whose exponent is algebraically the smallest (think, negative numbers are algebraically smaller than positive ones, and the "more negative" the number, the smaller information technology is).
If the exponents of the numbers are not the same, alter the number with the smaller exponent. Exercise this past moving the decimal point of the coefficient of that number to the left, and adding i to the exponent of that number, until the 2 exponents are equal.
Add together or subtract the coefficients of the two numbers. The outcome is the coefficient of the issue. The exponent is the exponent of the number you did not alter.
Put the issue in standard form, if necessary.

Examples:

a) (3 ten ten^-6) - (2 x ten^-7)

The algebraically smallest exponent is -7, so we change the 2d term:

2 10 10^-7 = 0.2 x x^-6 The exponents are at present the same

(3 x 10^-6) - (0.2 x ten^-6) = ( 3 - 0.two ) x 10^-6 = ii.viii ten 10^-vi

b) (ix.39 10 x⁵) + (8 x 10^three) = (9.39 10 ten^v) + (0.08 ten 10⁵) = 9.47 10 10^five

In situations where addition and subtraction are mixed with multiplication and division, practice the multiplication and division offset, and then do the addition and subtraction. And don't forget that raising things to powers always takes priority over multiplication and division!

Examples:

a) (5 x 10⁶) x (3 ten ten^-iii) + (two.ii ten 10⁵)

(five ten three) x (ten^vi x10^-3) + (ii.2 x 10^v)

(15 x 10^vi-3) + (2.2 x x⁵)

(xv x 10ⁱⁱⁱ) + (two.ii 10 x⁵)

(0.fifteen x x^v) + (2.2 ten 10^v)

2.4 x x⁵

b) (6.3 x ten³) - (4 x x⁴)³ / (8 x 10⁵)²

(6.three ten 10^three) - (4 x 4 x 4 10 ten^iv10 10⁴10 x⁴) / (8 x 8 x 10^fivex 10^five)

(six.3 x 10^three) - (64 10 x¹²) / (64 x 10¹⁰)

(6.3 x 10^three) - ( (64 / 64) x 10 ^{12 - 10}) )

(six.3 ten 10ⁱⁱⁱ) - (1 x 10²)

(6.3 x 10³) - (0.1 x 10³)

6.2 x 10³

Fourth Problem Set

Calculate the following:

1. (5.vii x 10⁶) + (3 x ten⁵)	two. (4.2 x 10^-viii) - (two.3 x 10^-viii)
iii. (3.8 x x⁵) - (2.i x 10^{half dozen})	4. (7.43 x 10^v) + (1.97 10 10⁷) / (2 x 10³)
5. (i.35 x 10⁷) + (8 x 10⁵)	half dozen. (half dozen.52 x 10^three) - (1.41 x 10⁵) x (2.31 x 10^-3)
7. (8.52 ten 10^-nine) + (2.16 x 10^-9)	eight. (4.73 x 10⁴) + (3.16 x 10¹¹) / (7.4 x tenⁱⁱⁱ)^two