ALGEBRAIC OPERATIONS Machinery's Handbook, 31st Edition
27
Algebraic Operations Algebraic operations rely on the symbols and properties of real numbers, reviewed below from ARITHMETIC and used throughout this section. Key skills include expo- nent use (for both powers and roots), polynomial operations, and solving equations for a variable (unknown). The same skills carry over to areas of geometry, trigonometry, and calculus, where a strong foundation in algebra is essential. Properties of Monomials and Exponents a ... a + + na = n terms ( ) a +0 a = a a – ( ) + = 0 a b × ab = a a ... a × × × a n = n factors ( ) a n a m a n m + ( ) = a n a m --- a n m – ( ) = a m ( ) n a mn = a m n / a 1 n / ( ) m = a n – 1 a n = --- 1 a n – ---- a n = a 0 1 (for a 0) ≠ = Examples: a a + 2 a = a 2 a 3 a 2+3 a 5 = = a 2 ( ) 3 a 2 3 × a 6 = = a –3 1 3 = = ab ( ) 2 a 2 b 2 = 3 4 3 a a × aa a 2 = = 12 12 a – a – a = – 3 4
1 a --
a 3 b 3 = ---
a 4 a 3 --- a 4 – 3 a = =
a b --
a 3 ---
Properties of Radicals
a b ----
a b = --
a a × a =
ab =
a b ×
n
m
a n m
a m n /
a n ( )
a n ( )
=
=
a =
, unless a < 0, m and n are both even
1 a n ---- a –1 n / = =
1 a -- n
a b + ( ) 2
a b + ( ) a b + ( ) a 2 ab b + + = =
Examples:
a 3 b 3 = ---- a 3 4
a b -- 3
3
a 3
a 3 ×
a 3 ×
a 3 ( )
ab 3
a 3
b 3
=
=
=
a
×
1 a 3 ----
1 a -- 3
2
1 a 1/3
a 3 2
a 3 ( )
a 2/3
a 1 ⁄ 3 –
=
=
= =
a 4 3
a 4 3 ×
=
=
=
( a 1/4 ) 1/3
=
x x x x x x x = = = 3 2 2
x + − = + − x 9 4
1 9 4 ( )
6
x = x 2
x
x
x
x = ( ) 5
3
5 3
5 3 /
3
x x 2 ˙ ˙ ˙ =
2
x 100 100 10 x = =
x
x
x
x
= x x x x 4 5 2 5
=
=
20
4 5
+ = ( ) x x x x x 4 5 3 5 ˙
= + = 125 45 125 45125 2 x x x x x x x x +
+
36 x 4 ----- 6 x 2 --- = =
36 x 4 ---
x + 1 cannot be simplified
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