
Review: Rules of Exponents
Rule 1: (a × b)^{n} = a^{n} × b^{n}
Rule 2: (a / b)^{n} = a^{n} / b^{n}
Rule 3: a^{ n} = 1 / a^{ n}
Rule 4: (a / b)^{ n} = (b / a)^{ n}

Simplify the expression 5^{ 2}
Solution
Use rule 3 above to write
5^{ 2} = 1 / 5^{ 2} = 1 / 25

(1/3)^{ 2} =
Solution
Rewrite expression as follows
(1/3)^{ 2} = [ (1)(1/3) ]^{ 2}
Use rule 1 of exponents
= (1)^{ 2} (1/3)^{ 2}
Use rules 3 and 4 of negative exponents
= [ 1 / (1)^{ 2} ] [ (3/1)^{ 2} ]
= [ 1/1 ] [ 9/1 ] = 9

Simplify the expression (5^{ 1}) / (3^{ 1})
Solution
Use rule 2 to rewrite expression as follows
(5^{ 1}) / (3^{ 1}) = (5 / 3)^{ 1}
Use rule 4 to rewrite expression as follows
= (3 / 5)^{ 1} = 3 / 5

(2^{ 3}) × (3^{ 2}) =
Solution
Use rule 3 to rewrite expression as follows
(2^{ 3}) × (3^{ 2}) = (1 / 2^{ 3}) × (1 / 3^{ 2}) = (1 / 8)(1 / 9) = 1 / 72

Simplify the expression  2^{ 3}
Solution
We first rewrite the given expression as follows
 2^{ 3} = (1)(2^{ 3})
We now use rule 3
= (1)(1 / 2^{ 3}) = (1)(1 / 8) =  1 / 8

 1^{ 3} + 2^{ 3} =
Solution
Use rule 3 to rewrite the given expression as follows
 1^{ 3} + 2^{ 3} = (1) (1 / 1^{ 3}) + 1 / 2^{ 3} =  1 / 1 + 1 / 8 =  1 + 1 / 8
Set fractions to common denominator
=  8 / 8 + 1 / 8 =  7 / 8

( 1)^{ 4} + 2^{ 3} =
Solution
Use rule 3 to rewrite the given expression as follows
( 1)^{ 4} + 2^{ 3} = 1 / (1)^{ 4} + 1 / 2^{ 3} = 1 / 1 + 1 / 8
Set fratcions to common denominator
= 8 / 8 + 1 / 8 = 9 / 8

Simplify the expression ( 4^{ 2})(2^{ 2})
Solution
Use rule 3 to rewrite the given expression as follows
( 4^{ 2})(2^{ 2}) =  (1 / 4^{ 2} )(2^{ 2})
Simplify
=  (1 / 16)(4) =  1 / 4

( 1)^{ 3} + 2^{ 0} =
Solution
Note that 2^{ 0} = 1. Use rule 3 to write
( 1)^{ 3} + 2^{ 0} = 1 / ( 1)^{ 3} + 1
Simplify
= 1 / (1) + 1 = 1 + 1 = 0

0^{ 3} =
Solution
Use rule 3 to write
0^{ 3} = 1 / 0^{ 3} = 1 / 0 : division by 0 not allowed and therefore 0^{ 3} is not a real number.
