Title 2 Simplifying Algebraic Expressions.pdf ✅

MSTC 2: SIMPLIFYING ALGEBRAIC EXPRESSIONS 1. Laws of Exponents Basic definition of exponents a n = a⏟ × a × a × ⋯ × a n times Derivation of Formulas: a m × a n = a m+n From the definition a m × a n = a⏟ × a × ⋯ × a m times × a⏟ × a × ⋯ × a n times Combine a⏟ × a × ⋯ × a m times × a⏟ × a × ⋯ × a n times = a⏟ × a × ⋯ × a (m+n)times From the definition a⏟ × a × a × ⋯ × a (m+n)times = a m+n Transitive Property of Equality a m × a n = a m+n a m a n = a m−n From the definition a m a n = a⏟ × a × a × ⋯ × a m times a⏟ × a × a × ⋯ × a n times Combine a⏟ × a × a × ⋯ × a m times a⏟ × a × a × ⋯ × a n times = a⏟ × a × a × ⋯ × a (m−n)times From the definition a⏟ × a × a × ⋯ × a (m−n)times = a m−n Transitive Property of Equality a m a n = a m−n
a 0 = 1; a ≠ 0 Change 0 to m − m a 0 = a m−m From a m a n = a m−n a m−m = a m am Simplify a m am = 1 Transitive Property of Equality a 0 = 1 a −n = 1 a n Change −n to 0 − n a −n = a 0−n From a m a n = a m−n a 0−n = a 0 a n Since a 0 = 1 a 0 a n = 1 a n Transitive Property of Equality a −n = 1 a n (ab)m = a mb m From the definition (ab)m = ab⏟ × ab × ab × ⋯ × ab m times Rearrange the factors ab⏟ × ab × ab × ⋯ × ab m times = a⏟ × a × ⋯ × a m times × b⏟ × b × ⋯ × b m times From the definition a⏟ × a × ⋯ × a m times × b⏟ × b × ⋯ × b m times = a mb m Transitive Property of Equality (ab)m = a mb m
( a b ) m = a m bm From the definition ( a b ) m = a b × a b × a b × ⋯ × a ⏟ b m times Multiply the numerators and denominators a b × a b × a b × ⋯ × a ⏟ b m times = a⏟ × a × a × ⋯ × a m times b⏟ × b × b × ⋯ × b m times From the definition a⏟ × a × a × ⋯ × a m times b⏟ × b × b × ⋯ × b m times = a m bm Transitive Property of Equality ( a b ) m = a m bm (a m) n = a mn From the definition (a m) n = a m × a m × a m × ⋯ × a ⏟ m n times From a m × a n = a m+n a m × a m × a m × ⋯ × a ⏟ m n times = a mn Transitive Property of Equality (a m) n = a mn Compilation of Derived Formulas: a ma n = a m+n a −n = 1 a n (ab)m = a mb m (a m) n = a mn a m a n = a m−n a 0 = 1; a ≠ 0 ( a b ) m = a m bm