Properties of exponents
Properties Examples
1) a a a = an,  n factors (exponent), a -base,   a3 = a a a,      24 = 2 2 2 2 = 16,
2)   a = a1,  or   a1 = a,   2 = 21,    (-3)1 = -3,
3)   a0 = 1,   an  an = 1,   23 23  = 23 -= 20  = 1,     (ab3)0 = 1,
4)   1n  = 1,   15 = 1,      (-1)5 = -1,
5)   (- a)2n = a2n,    2n -even exponents,   (- 5)4 = 54 = 625,
6)   (- a)2n -1 = - a2n -1,   2n-1 -odd exponents,   (- 2)3 = - 23  = - 8,
7)   (-1)2n = 1,   (-1)6 = 1,
8)   (-1)2n -1 = -1,     (-1)7 = -1.
The rules for powers or exponents
Rules Examples
1)   am a = am + n,   a)  a3 a4 = a3 + 4,                 b)  24 25 = 24 + 5 = 29 = 512,
2)   am a = am - n,   a)  x5 x3  = x5 - 3  = x2,       b)  0.14 0.13 = 0.14 - 3 = 0.1, 
3)   an b = (a b)n,   a)  24 34 = (2 3)4 = 64,        b)  45 0.85 = (4 0.8)5 = 3.25,
4)   an b = (a b)n,   a)  x6 y6  = (x y)6,             b)  125 35 = (12 3)5 = 45,
5)    
6)   (am) = am n,   a)  (a3)4 = a3 4 = a12,             b)  (45)3 = 45 3 = 415,
7)    
8)    
9)    
Simplifying an exponential expression
Use the above rules of powers (or exponential laws) and the rules of algebra to simplify expressions with numerical and variable bases, as show examples:
Examples:
Order of operations
When expressions have more than one operation, we have to follow rules for the order of operations, i.e., the order in which to evaluate a mathematical expression:
1.  First do all operations that lie inside parentheses, if there are parentheses inside parentheses work from the innermost parentheses out.
Anything that acts as a grouping symbol, like any expression in the numerator or denominator of a fraction or in an exponent is also considered grouped and should be simplified before carrying out further operations.
2.  Next, do any work with exponents or radicals.
3.  Then, working from left to right, do all multiplication and division.
4.  And finally, working from left to right, do all addition and subtraction.
Example: - 8 -  3 (- 4) + 15 (- 5) = - 8 - (- 12) + (- 3) = - 8 + 12 - 3 = 1
Example: - 3 + 2 [- 2 - 3 (- 4 2 + 1)] = - 3 + 2 [- 2 - 3 (- 2 + 1)] = - 3 + 2 [- 2 - 3 (- 1)]
                                                = - 3 + 2 [- 2 + 3] = - 3 + 2 = - 1
Example: Find the value of the expression when x = - 0.5 and y = - 1.
Solution: Substituting  x = - 0.5 and y = - 1 gives
College algebra contents
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