Properties of Exponent Numbers

Solving on the problems of the exponent numbers is how to apply properties that exist in exponent numbers. Therefore, to be able to solve on exponent numbers problem we must understand the definition of exponent numbers and know how to show the properties of the exponent numbers so that memorizing its properties becomes easier.

Definition of Exponent Numbers
The following positive integer numbers ($ n \in Z^+ $), $ a $ is called the base number or principal number and $ n $ is called the exponent defined as follows.

$ a^n = \underbrace {a \times a \times a \times ... \times a}_{\mbox{n factor}} $; $ a \in R $

Whereas the negative exponent is defined as follows.

$ \begin{align} a^{- m} & = (\frac{1}{a})^m \\ & = \frac{1}{a^m} \end{align} $; $ a \in R $

Example:
$ 2^3 = 2 \times 2 \times 2 = 8 $
$ (- 4)^3 = (- 4) \times (-4) \times (-4) = - 64 $
$ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $
Zero Exponent
Not all real numbers raised by 0 are 1. However, there are exceptions that this does not apply to 0. So, for every $ a \in R $; $ a \neq 0 $ applies:

$ a^0 = 1 $

Evidence:
$ \begin{align} \frac{a^n}{a^n} & = 1 \\ \Leftrightarrow a^{n-n} & = 1 \\ \Leftrightarrow a^{0} & = 1 \end{align} $
Characteristics of Positive Exponent
If $ a \in R $, m and n are natural numbers then

$ a^m \times a^n = a^{m + n} $
$ a^m:a^n = a^{m-n} $
$ (a^m)^n = a^{mn} $
$ (a \times b)^n = a^n b^n $
$ (\frac{a}{b})^n = \frac{a^n}{b^n}; \ b \neq 0 $

Characteristics of Fraction Exponent

$ (a^{\frac{m}{n}})(a^{\frac{p}{n}}) = a^{\frac{m + p}{n}} $
$(a^{\frac{m}{n}})(a^{\frac{p}{q}}) = a^{\frac{m}{n} + \frac{p}{q}} $

Sample Questions
Examples of school questions:
1. $ 2^2 \times 2^5 = 2^{2 + 5} = 2^{7} $
2. $ (2^3)^2 = 2^6 $
3. $ (\frac{2}{3})^3 = \frac{2^3}{3^3} = \frac{8}{27} $

The following numbers whose biggest value is ...
a. 777
b. $ 7^{77} $
c. $ (77)^7 $
d. $ (7^7)^7 $
e. $ (7 \times 7)^7 $
Answer B
We use the properties of a number to answer the question.
a. $ 777 = 7(111) <7(11^2) $
b. $ 7^{77} = 7^{57} 7^{18} 7^2 = 7^{57} 7^{18} 49 $
c. $ 77^7 = (7 \times 11)^7 = 7^7 \times 11^7 = 7^7 11^611 $
d. $ (7^7)^7 = 7^{49} $
e. $ (7 \times 7)^7 = (7^2)^7 = 7^{14} $

Note that $ 7^{77} > 7^{49} > 7^{14}$, therefore $ b > d > e $. Because $ 7^3 > 11^2 $ then $ 7^{57} 7^{18} 49 > 7^7 11^6 11 > 7 \times 11^2 > 7(111) $, so $ b > c> a $.

Examples of Olympic questions: Determine the unit number of $ 7^{1234}$
Solution:
Consider the following pattern!
$ \begin{align} 7^1 & = 7 \\ 7^{2} & = ... 9 \\ 7^{3} & = ... 3 \\ 7^{4} & = ... 1 \\ 7^{ 5} & = ... 7 \\ 7^{6} & = ... 9 \\ 7^{7} & = ... 3 \\ 7^{8} & = ... 1 \end{align} $
Based on this pattern and using the properties of the Exponent numbers obtained:
$ \begin{align} 7^{1234} & = 7^{(4 \times 308)} \times 7^2 \\ & = (7^4)^{308} \times 7^2 \end{align}$.
Because units of $ 7^4 $ are 1 and $ 7^2 $ is 9, the unit number of $ 7^{1234} $ is $ 1 \times 9 = 9 $.

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