In working problems in astronomy, you will often encounter very big or very
small numbers. For example, the number of stars in our galaxy the Milky Way
is about 100 billion. Large numbers like 100 billion are cumbersome to write
out in ordinary form (100,000,000,000) and so scientists use a shorthand
notation called powers of ten.
In powers of ten notation, large numbers are written using ten to a
power, or exponent. The exponent tells you how many times ten is to be
multiplied by iteslf to equal the number you wish to write. For example,
100 can be written as 10x10 = 102. 10,000 = 10x10x10x10
= 104. Notice that the exponent of the ten is simply the number of
1.What is 100 billion =100,000,000,000 in powers of ten notation?
Many people get confused about how to write 10 and 1 in this form. 10 =
101 and 1 = 100.
Very small numbers can also be written in powers of ten notation, but small
numbers use negative exponents. For example, 0.01 = 10-2. Why
the minus sign? It is easy to see why if you first write 0.01 as 1/100 =
1/102. Then it becomes clear that the negative power simply
indicates the power of ten of the denominator when you have written the
number as a fraction.
Note: An easy way to figure out the power of ten for small numbers is to
count the zeros after the decimal and add one.
2. What is 0.0001 in powers of ten notation?
3.What is this in powers of ten notation?
Here the trick is to write the number as the non-zero digits times a power of
ten. For example, 475,000,000=4.75x100,000,000 = 4.75x108.
Similarly, 0.00031 = 3.1x0.0001 = 3.1x10-4.
Notice in the above that you could also have written 475,000,000 as
475x106 or 47.5 x107. It is generally easier to read
the number if you write only one digit to the left of the decimal point, but
use your judgement.
4. What is 31,600,000 in powers of ten notation?
The rules are simple.
To Multiple, add the exponents.
To Divide, subtract the exponents.
For example to multiply: 106x104 = 106+4 =
To divide: 105/103 = 105-3 = 102.
5. What is 103x105?
7. What is 3.16x107 x 2.99x108?
As another example (106)1/2 = 106x(1/2) =
Note that raising a number to a fractional power is equivalent to taking a root
of the number. In particular, the 1/2 power = the square root; the 1/3 power
= the cube root, and so on. Thus, in the last example we took the square
root of one million = (106).
You must be careful when raising numbers to powers to take all the
numbers involved to the power. For example, (3.14x24x103)3 =
31.0x1.38x104x103x3 = 4.28x101+4+9 =
8. What is (2x103)2?
Whew! Had enough?