5.14 × 10

Scientific notation consists of a *coefficient* (here 5.14)
multiplied by 10 raised to an *exponent* (here 5). To convert
to a real number, start with the base and multiply by 5 tens like
this: 5.14 × 10 × 10 × 10 × 10 × 10 =
514000.0. Multiplying by tens is easy: one simply moves the decimal
point in the base (5.14) 5 places to the *right*, adding extra
zeroes as needed.

5.14 × 10^{5}

= 51.4 × 10^{4}

= 514.0 × 10^{3}

= 5140.0 × 10^{2}

= 51400.0 × 10^{1}

= 514000.0 × 10^{0}

= 514000.0

= 51.4 × 10

= 514.0 × 10

= 5140.0 × 10

= 51400.0 × 10

= 514000.0 × 10

= 514000.0

0.000345 = 3.45 × 10

Here we wish to write the number 0.000345 as a *coefficient*
times 10 raised to an *exponent*. To convert to scientific
notation, start by moving the decimal place in the number until you
have a *coefficient* between 1 and 10; here it is 3.45. The
number of places to the *left* that you had to move the decimal
point is the exponent. Here, we had to move the decimal 4 places to
the right, so the exponent is -4.

0.000345

= 0.00345 / 10

= 0.0345 / (10 × 10)

= 0.345 / (10 × 10 × 10)

= 3.45 / (10 × 10 × 10 × 10)

= 3.45 / (10^{4})

= 3.45 × 10^{-4}

= 0.00345 / 10

= 0.0345 / (10 × 10)

= 0.345 / (10 × 10 × 10)

= 3.45 / (10 × 10 × 10 × 10)

= 3.45 / (10

= 3.45 × 10

(9 × 10

Multiplications and divisions can be done in any order - take
advantage of this! First, multiply the two coefficients and then
multiply the two powers of ten by adding their exponents: since -1 +
10 = 9, then 10^{-1} × 10^{10} =
10^{9}. Finally, combine your two answers and convert to
scientific notation: 27 × 10^{9} = 2.7 ×
10^{10}. In symbols:

(9 × 10^{-1}) × (3 × 10^{10})

= (9 × 3) × (10^{-1} × 10^{10})

= (27) × (10^{9})

= 2.7 × 10^{10}

= (9 × 3) × (10

= (27) × (10

= 2.7 × 10

(3.5 × 10

Distribute the division across both the coefficients and the powers of
ten. Next, divide the two coefficients: 3.5/5 = (35/10)/5 = (35/5)/10
= 7/10 = 0.7. Then, divide the two powers of ten by subtracting their
exponents: since -6 - (-2) = -6 + 2 = -4, then 10^{-6} /
10^{-2} = 10^{-4}. Finally, combine your two answers
and convert to scientific notation. In symbols:

(3.5 × 10^{-6}) / (5 × 10^{-2})

= (3.5 / 5) × (10^{-6} / 10^{-2})

= (0.7) × (10^{-4})

= 7 × 10^{-5}

= (3.5 / 5) × (10

= (0.7) × (10

= 7 × 10

4.9 × 10

First, factor out one of the powers of ten; either will work, but the smaller one may be easiest. This involves dividing both numbers by the power of ten and multiplying the whole quantity by the same power of ten. To divide one power of ten by another, simply subtract the two exponents (see Multiplication/Division). Next, convert the two numbers from scientific notation to real numbers. Now add the two numbers normally. Finally convert to scientific notation if the coefficient is less than 1 or greater than 10.

4.9 × 10^{2} + 7.9 × 10^{3}

= (4.9 × 10^{2}/10^{2} + 7.9 × 10^{3}/10^{2} ) × 10^{2}

= (4.9 × 10^{0} + 7.9 × 10^{1}) × 10^{2}

= (4.9 + 79) × 10^{2}

= 83.9 × 10^{2}

= 8.39 × 10^{3}

= (4.9 × 10

= (4.9 × 10

= (4.9 + 79) × 10

= 83.9 × 10

= 8.39 × 10

Another way to perform any operation on two scientific notation numbers is to convert both to normal numbers, then perform the operation and finally convert the result back to scientific notation. This method is cumbersome, however, if either exponent is very large or very small. Here it works beautifully.

4.9 × 10^{2} + 7.9 × 10^{3}

= 490 + 7900

= 8390

= 8.39 × 10^{3}

= 490 + 7900

= 8390

= 8.39 × 10

4.9 × 10

As with addition, start by factoring out one of the powers of ten. Next, convert both scientific notation numbers to real numbers. Subtract the two numbers normally and convert to scientific notation if the coefficient is not between 1 and 10 (or -1 and -10).

4.9 × 10^{-6} - 7.9 × 10^{-5}

= (4.9 × 10^{-6} / 10^{-6} - 7.9 × 10^{-5} / 10^{-6} ) × 10^{-6}

= (4.9 × 10^{0} - 7.9 × 10^{1}) × 10^{-6}

= (4.9 - 79) × 10^{-6}

= - (79 - 4.9) × 10^{-6}

= -74.1 × 10^{-6}

= -7.41 × 10^{-5}

= (4.9 × 10

= (4.9 × 10

= (4.9 - 79) × 10

= - (79 - 4.9) × 10

= -74.1 × 10

= -7.41 × 10