When you have a sum of several equal terms, you introduce
their product
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 2 · 8. For products
of equal factors, you use powers:
| 10 · 10 · 10 · 10 · 10 · 10 ·10 · 10 · 10 · 10 · 10 · 10 = 1012. |
The index 12 tells the number of the factors 10 in the product. Other examples are:
| 63=6·6·6 | 414=4·4·4·4·4·4·4·4·4·4·4·4·4·4 | a4=a·a·a·a |
They are the third, fourteenths and fourth powers of 6, 4, a.
Definition
The power an is the product of n factors of the base a.
This is very convenient for large and small numbers:
| The Sun's surface is 6·1012 km2, | the Sun's mass is 6.1·1024 kg, | 0.000 000 5 = 5/107. |
The exponents n of 10 give 1 million (n = 6), 1 billion (n = 9), 1 trillion (n = 12), 1 quadrillion (n = 15). . .
If you have to multiply or divide powers, you must inspect carefully their structure:
| 1. a3·a2=a5 | 2. a2·a4=a6 | 3. a5/a3=a2 |
Examples
| 1. 2a2·4b·4a = 2·4·4·a2·a·b = 32a3·b | 2. 6b2·4b·3a= 3·4·6·ab2b = 72ab3 | |
| 3. ab2c/bc = ab | 4. 6a2b2c/3ac = 2ab2 |
You can only simplify expressions with terms with the same base, but not a2 + a or a2b + ab!
| 5. 2a2+5a2=7a2 | 6. 4a2+3b-2a2+5b=2a2+8b | 7. 6a2b+2ab2-4a2b - ab2+ abc=2a2b+ab2+abc |