# Scientific Notation

## Conversion Problems

### Convert from Scientific Notation to Real Number:5.14 × 105 = 514000.0

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 × 105
= 51.4 × 104
= 514.0 × 103
= 5140.0 × 102
= 51400.0 × 101
= 514000.0 × 100
= 514000.0

### Convert from Real Number to Scientific Notation:0.000345 = 3.45 × 10-4

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 / (104)
= 3.45 × 10-4

## Multiplication/Division Problems

### Multiply Two Numbers Written in Scientific Notation:(9 × 10-1) × (3 × 1010) = 2.7 × 1010

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 × 1010 = 109. Finally, combine your two answers and convert to scientific notation: 27 × 109 = 2.7 × 1010. In symbols:

(9 × 10-1) × (3 × 1010)
= (9 × 3) × (10-1 × 1010)
= (27) × (109)
= 2.7 × 1010

### Divide Two Numbers Written in Scientific Notation:(3.5 × 10-6) / (5 × 10-2) = 7 × 10-5

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

### Add Two Numbers Written in Scientific Notation:4.9 × 102 + 7.9 × 103 = 8.39 × 103

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 × 102 + 7.9 × 103
= (4.9 × 102/102 + 7.9 × 103/102 ) × 102
= (4.9 × 100 + 7.9 × 101) × 102
= (4.9 + 79) × 102
= 83.9 × 102
= 8.39 × 103

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 × 102 + 7.9 × 103
= 490 + 7900
= 8390
= 8.39 × 103

### Subtract Two Numbers Written in Scientific Notation:4.9 × 10-6 - 7.9 × 10-5 = -7.41 × 10-5

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 × 100 - 7.9 × 101) × 10-6
= (4.9 - 79) × 10-6
= - (79 - 4.9) × 10-6
= -74.1 × 10-6
= -7.41 × 10-5