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# Scientific Notation Rules for Physics Help (page 2)

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By McGraw-Hill Professional
Updated on Sep 17, 2011

## Division

When numbers are divided in power-of-10 notation, the decimal numbers (to the left of the multiplication symbol) are divided by each other. Then the powers of 10 are subtracted. Finally, the quotient is reduced to standard form. Let’s go another round with the same three number pairs we’ve been using:

(3.045 × 10 5 )/(6.853 × 10 6 ) = (3.045/6.853) × (10 5 /10 6 )

≈ 0.444331 × 10 (5−6)

= 0.444331 × 10 −1

= 0.0444331

(3.045 × 10 −4 )/(6.853 × 10 −7 ) = (3.045/6.853) × (10 −4 /10 −7 )

≈ 0.444331 × 10 [−4−(−7)]

= 0.444331 × 10 3

= 4.44331 × 10 2

= 444.331

(3.045 × 10 5 )/(6.853 × 10 −7 ) = (3.045/6.853) × (10 5 /10 −7 )

≈ 0.444331 × 10 [5−(−7)]

= 0.444331 × 10 12

= 4.44331 × 10 11

Note the “approximately equal to” sign (≈) in the preceding equations. The quotient here doesn’t divide out neatly to produce a resultant with a reasonable number of digits. To this, you might naturally ask, “How many digits is reasonable?” The answer lies in the method scientists use to determine significant figures. An explanation of this is coming up soon.

## Exponentiation

When a number is raised to a power in scientific notation, both the coefficient and the power of 10 itself must be raised to that power and the result multiplied. Consider this example:

(4.33 × 10 5 ) 3 = (4.33) 3 × (10 5 ) 3

= 81.182737 × 10 (5×3)

= 81.182737 × 10 15

= 8.1182727 × 10 16

If you are a mathematical purist, you will notice gratuitous parentheses in the first and second lines here. From the point of view of a practical scientist, it is more important that the result of a calculation be correct than that the expression be as mathematically lean as possible.

Let’s consider another example, in which the exponent is negative:

(5.27 × 10 −4 ) 2 = (5.27) 2 × (10 −4 ) 2

= 27.7729 × 10 (−4×2)

= 27.7729 × 10 −8

= 2.77729 × 10 −7

## Taking Roots

To find the root of a number in power-of-10 notation, the easiest thing to do is to consider that the root is a fractional exponent. The square root is the same thing as the 1/2 power; the cube root is the same thing as the 1/3 power. Then you can multiply things out in exactly the same way as you would with whole-number powers. Here is an example:

(5.27 × 10 −4 ) 1/2 = (5.27) 1/2 × (10 −4 ) 1/2

≈ 2.2956 × 10 [(−4×l/2)]

= 2.2956 × 10 −2

= 0.02956

Note the “approximately equal to” sign in the second line. The square root of 5.27 is an irrational number, and the best we can do is to approximate its decimal expansion. Note also that because the exponent in the resultant is within the limits for which we can write the number out in plain decimal form, we have done so, getting rid of the power of 10.

Practice problems for these concepts can be found at: Scientific Notation for Physics Practice Test

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