# Powers of Ten and Calculations

Power of ten is the mathematical shorthand for writing
very large or very small numbers, using positive

or negative exponents of base 10. The exponent is written as a superscript
numeral to the right of the 10.

**The Basics:**

(1) Ten with a positive exponent means to multiply 10 by itself the number of
times indicated by the

power. For example, 10^{3} = 10 x 10 x 10 = 1000.

(2) Ten to a negative power means to multiply 0.1 (1/10) by itself the number of
times indicated by

the power. For example, 10^{-3} = 0.1 x 0.1 x 0.1 = 0.001 (equivalent to 1/1000, or
the reciprocal of

10^{3}, which is 10^{-3}).

(3) Ten to the zero power (10^{0}) equals 1, by definition. Zero (0) can only be
represented by itself.

(4) Any number can be expressed as a power of ten by using a scalar value and an
exponent. For

example, 212 = 2.12 x 100 = 2.12 x 10^{2} and 0.0098 = 9.8 x 10^{-3}.

(5) When multiplying two numbers expressed as powers of ten, add the exponents.
For example,

(6) When dividing two numbers expressed as powers of ten, subtract the
exponents. For example,

and

(7) When multiplying or dividing numbers in scientific notation, multiply (or
divide) the scalar

values separately from the exponents, then combine the two.

For example,

(8) When adding or subtracting numbers expressed in powers of ten, transform the
values to a

common exponent, then add or subtract the scalar (non-exponent) values. For
example, 3.8 x 10^{3}

To make thinking in large and small numbers a little easier, scientists and
engineers commonly

express powers of ten in multiples of 3 (or –3), which is equivalent to
counting, 1 thousand, 1 million, 1

billion, 1 trillion, and so on. In computer lingo, this idea is expressed by the
prefixes kilo-, mega-, giga-,

and tera-, as in “megabytes.”

Equivalent value | Equivalent value | ||

**Other things to keep in mind about math with exponents:**

Use a leading zero in the ones place for any decimal
values

for example, 0.025 instead of .025

When using scientific notation, express the scalar value with a numeral in the
ones place

for example, 1.25 x 10^{5} instead of 0.0125 x 10^{7}.

The exception to the above guideline is when you need to use a common exponent
for

operations such as addition and subtraction;

for example,

instead of

Be careful about moving the decimal point and the exponent in the correct
direction

(especially for negative exponents). In your mind (or on paper),
convert the number

expressed in scientific notation into a decimal value, and
ask yourself whether the

exponent gets larger or smaller. (Admittedly, this can
be tricky.)

For example,

0.254 x 10^{2} (0.254 x 100 = 25.4) becomes 2.54 x 10^{1}, NOT 2.54 x 10^{3} (which is
2540)

and

25.4 x 10^{2} (25.4 x 100 = 2540) becomes 2.54 x 10^{3}, NOT 2.54 x 10^{1} (which is
25.4)

Likewise,

0.15 is equivalent to 1.5 x 10^{-1} or 0.015 x 10^{1} (moving the decimal point either
way)

**A rule of thumb for decimal places and exponents:**

For positive exponents, 1 x 10^{n} n is equal to the number of zeroes to the right
of 1.

Examples:

For negative exponents, 1 x 10^{-n} there are (n-1) zeroes to the left of 1

(i.e., the equivalent decimal position would be to the right of the 1)

Examples:

Remember the relationship between positive and negative exponents and
reciprocals of

fractions:

and

AND, when adding or subtracting with scientific notation, always remember to
convert to

a common exponent:

which is equivalent to |
||||

convert | ||||

to |

Express each number as a power of ten:

(can zero be represented as a power of ten?)

Express each as a non-“power of ten” number:

Complete each calculation, show intermediate steps when
appropriate, and express the

answer as a power of ten: