6.5 Rules of Powers
There are rules which you can use in order to simplify calculations with powers. It is important though that you understand the rules before you use them. In order to understand these rules, you should first have gone through and understood chapter 6.1  6.4.
1. Powers with the same base which are multiplied with each other.
Example:
4^{2 }· 4^{3} = (4 · 4) · (4 · 4 · 4) =
4^{2+3} = 4^{5}
7^{4 }· 7^{8} = 7^{4+8} =
7^{12}
5^{2 }· 5^{1} = 5^{2+(1)} = 5^{21} =
5^{1}
2. Powers with the same base that are divided with each other.
Example
:
a)
6^{5} ÷ 6^{2} = (6 · 6 · 6 · 6 · 6) ÷ (6 · 6) =
6^{52} = 6^{3}
b) 5^{3} ÷ 5^{5} = 5^{35} = 5^{2} = 1 ÷ 5^{2 }
c) 2^{3} ÷ 2^{3} =
2^{33} = 2^{0} = 1
3. Powers with the same exponent that are multiplied by each other.
Example:
2^{4 }· 3^{4} = (2 · 3)^{4} = 6^{4}
6 ^{12 }· 3^{12} = (6 · 3)^{12} =
18^{12}
4^{5 }· 7^{5} = (4 · 7)^{5} = 28^{5}
4. Powers with the same exponent which are divided by each other.
Example:
6^{2} ÷ 2^{2} = (6 ÷
2)^{2} = 3^{2}
8^{3} ÷ 4^{3} = (8
÷ 4)^{3} = 2^{3 }
9^{4} ÷ 3^{4} = (9 ÷ 3)^{4} = 3^{4}
5. Powers with double exponents, in other words where the base is also in exponential form. You can even call it a “power of a power”.
Example:
(3^{2})^{5} =
3^{2 }· 3^{2 }· 3^{2 } · 3^{2 }· 3^{2} =
3^{2+2+2+2+2} = 3^{2·5} = 3^{10}
(4^{6})^{7} = 3^{6·7}=
3^{42
}(3^{4})^{2} =
3^{4·(2)}= 3^{8}
