# Powers of Ten Tutorial

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 the zeros.

1.What is 100 billion =100,000,000,000 in powers of ten notation?
 A).. 106 B)..108 C).. 109 D)..1011 E)..1012

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.

For example, 2. What is 0.0001 in powers of ten notation?
 A)... 103. B)...10-4 C)...10-3 D)...10-2 E)...10-1

As another example, the diameter of a hydrogen atom is roughly 0.0000000001 meters.

3.What is this in powers of ten notation?
 A)... 10-12. B)... 10-10. C)... 10-11. D)... 101/12. E)... 10-13.

How do we use powers of ten notation to write numbers like 475,000,000 or 0.00031?

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?
 A)... 3.16 x105 B)... 3.16 x106 C)... 3.16 x107 D)... 3.16 x108 E)... 3.16 x109

Now that you have some idea of how to write numbers in powers of ten notation, we should look at how to multiply and divide such numbers.

The rules are simple.

To Divide, subtract the exponents.

For example to multiply: 106x104 = 106+4 = 1010.

To divide: 105/103 = 105-3 = 102.

5. What is 103x105?
A)...?
 A)... 1021 B)...1010 C)...10-4 D)... 104 E)... 102.3

Let's do a slightly more complicated problem now.

7. What is 3.16x107 x 2.99x108?
 A)... 6.15x 1056 B)... 9.45x1015 C)... 9.45x10-1 D)... 9.45x101 E)...6.15x1015

So far we have discussed only multiplication and division using powers of ten. Sometimes you will need to raise numbers to a power. In that case, you multiply the exponents. For example, (105)3 = 105x3 = 1015.

As another example (106)1/2 = 106x(1/2) = 103.

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 = (3.143x243x(103)3 = 31.0x1.38x104x103x3 = 4.28x101+4+9 = 4.28x1014.

8. What is (2x103)2?
 A)... 2x105 B)...4x105 C)...4x106 D)... 2x106 E)... 4x109