Chapter 2 began with a single repeated multiplication and ended with a complete toolkit for moving exponents across any expression. Every section added one idea, and each idea was forced by the same logic. An exponent counts factors. Once that picture is clear, every law becomes something you could derive on your own. The pages below collect what the chapter built, law by law.
Key idea Squares and cubes. The square of b b b 2 = b × b b 2 = b × b b 3 = b × b × b b 3 = b × b × b b 2 b 2 b b b b b 3 b 3 b b b b b b 0 , 1 , 4 , 9 , 16 , 25 , … 0 , 1 , 4 , 9 , 16 , 25 , … 0 , 1 , 8 , 27 , 64 , 125 , … 0 , 1 , 8 , 27 , 64 , 125 , … n n n n 2 n + 1 2 n + 1 ( n + 1 ) 2 = n 2 + 2 n + 1 ( n + 1 ) 2 = n 2 + 2 n + 1 ( n − 1 ) 2 = n 2 − 2 n + 1 ( n − 1 ) 2 = n 2 − 2 n + 1
Key idea Signs and parentheses. An exponent applies only to the base written directly beneath it, unless parentheses make more of the expression the base. So − a n = − ( a n ) − a n = − ( a n ) ( − a ) n ( − a ) n ( − a ) 2 = a 2 ( − a ) 2 = a 2 ( − a ) 3 = − a 3 ( − a ) 3 = − a 3
Key idea Powers distribute over multiplication and division, not addition. ( a b ) n = a n × b n ( a b ) n = a n b n ( ab ) n = a n × b n ( b a ) n = b n a n ( a b ) n ( ab ) n n n a ⋅ b a ⋅ b a a b b ( a + b ) n ≠ a n + b n ( a + b ) n = a n + b n a a b b
The next three laws describe what happens when exponents meet multiplication, division, and repeated powering. All three require the same base throughout.
Key idea Three laws for a common base. a m × a n = a m + n (product rule) a m × a n = a m + n (product rule) a m a n = a m − n ( a ≠ 0 ) (quotient rule) a n a m = a m − n ( a = 0 ) (quotient rule) ( a m ) n = a m × n (power rule) ( a m ) n = a m × n (power rule)
Key idea The zero power. For any nonzero base a a a 0 = 1. a 0 = 1. a a a 1 = a a 1 = a a 0 a 0 a ÷ a = 1 a ÷ a = 1 a n a n a n a n a n − n = a 0 a n − n = a 0 1 1
Key idea What the exponent applies to. A zero power applies only to its own base, not to the whole expression. In c ⋅ b 0 c ⋅ b 0 b b c ⋅ b 0 = c × 1 = c c ⋅ b 0 = c × 1 = c ( c b ) 0 = 1 ( c b ) 0 = 1 − a 0 = − ( a 0 ) = − 1 − a 0 = − ( a 0 ) = − 1 ( − a ) 0 = 1 ( − a ) 0 = 1
The power ladder does not stop at zero. Carrying the dividing pattern one step further below zero lands on a reciprocal, and that one observation extends all three laws into the negatives without requiring any new rules.
Key idea Negative exponents. For any nonzero base a a a − n = 1 a n . a − n = a n 1 . a m × a − n a m × a − n a m − n a m − n a − n a − n a m a − n = a m + n a − n a m = a m + n ( 1 a ) n = a − n . ( a 1 ) n = a − n .
Key idea Stacked exponents and coefficients. Three or more stacked exponents collapse by multiplying every exponent together. ( ( a m ) n ) p = a m × n × p . ( ( a m ) n ) p = a m × n × p . ( c x m ) n = c n x m n . ( c x m ) n = c n x mn . ( 2 x 3 ) 4 = 16 x 12 ( 2 x 3 ) 4 = 16 x 12 2 x 12 2 x 12 ( 5 3 ) 2 = 5 6 ( 5 3 ) 2 = 5 6 5 3 × 5 2 = 5 5 5 3 × 5 2 = 5 5
Key idea A negative outer exponent gives the reciprocal. A fraction raised to a negative power equals its reciprocal raised to the positive power. ( a b ) − n = ( b a ) n = b n a n ( a ≠ 0 , b ≠ 0 ) . ( b a ) − n = ( a b ) n = a n b n ( a = 0 , b = 0 ) .
Key idea The common base strategy. When every base in an expression is a power of the same number, rewrite them all in that base first. With one base throughout, the three laws apply directly and the expression becomes a single power.
Every rule in Chapter 2 restates one idea. An exponent counts factors, and the laws describe how that count changes under each operation. Multiplication joins two counts, so the exponents add. Division cancels factors, so they subtract. Repeated powering makes copies of copies, so the exponents multiply. A zero exponent is a count of zero, an empty product, worth 1 1
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