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 \(b\) is \(b^2 = b \times b\) and the cube is \(b^3 = b \times b \times b\). Geometrically, \(b^2\) counts the cells in a \(b\)-by-\(b\) grid and \(b^3\) counts the unit cubes in a \(b\)-by-\(b\)-by-\(b\) solid. The perfect squares are \(0, 1, 4, 9, 16, 25, \ldots\) and the perfect cubes are \(0, 1, 8, 27, 64, 125, \ldots\) Each step up the staircase of perfect squares adds the next odd number, because growing an \(n\)-by-\(n\) square by one on each side wraps a strip of \(2n + 1\) new cells around the outside. That is the identity \((n+1)^2 = n^2 + 2n + 1\), and it runs backward too, \((n-1)^2 = n^2 - 2n + 1\).
Key idea The sign trap. An exponent grips only the base written directly beneath it, unless parentheses gather more into that base. So \(-a^n = -(a^n)\), not \((-a)^n\). An even exponent always produces a positive result because negative factors cancel in pairs, \((-a)^2 = a^2\). An odd exponent keeps the sign of its base, \((-a)^3 = -a^3\). The parentheses are the entire difference.
Key idea Powers distribute over multiplication and division, not addition. $$(ab)^n = a^n \times b^n \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$ Because \((ab)^n\) means \(n\) copies of the pair \(a \cdot b\), you can regroup to gather all the \(a\)-factors together and all the \(b\)-factors together. A quotient works the same way. But \((a + b)^n \ne a^n + b^n\) in general. A sum cannot be ungrouped the same way because \(a\) and \(b\) are not separate multiplicative factors of the expression.
The next three laws describe what happens when exponents meet the three fundamental operations: multiplication, division, and repeated powering. All three require the same base throughout.
Key idea Three laws for a common base. $$a^m \times a^n = a^{m+n} \quad \text{(product rule)}$$ $$\frac{a^m}{a^n} = a^{m-n} \quad (a \ne 0) \quad \text{(quotient rule)}$$ $$(a^m)^n = a^{m \times n} \quad \text{(power rule)}$$ The product rule fuses two separate runs of identical factors into one longer run. The quotient rule counts the factors that survive after equal ones cancel. The power rule counts how many factors accumulate when each copy of a power is expanded. None of the three laws apply across different bases.
Key idea The zero power. For any nonzero base \(a\), $$a^0 = 1.$$ Two roads arrive here. Walking down the power ladder, each step divides by \(a\), so the step from \(a^1 = a\) down to \(a^0\) gives \(a \div a = 1\). Alternatively, the quotient rule applied to \(\dfrac{a^n}{a^n}\) gives \(a^{n-n} = a^0\), and a nonzero quantity over itself is always \(1\). The zero power is not a separate rule bolted on. It is the only value that keeps the quotient rule honest.
Key idea What the exponent grips. The zero power collapses exactly what it touches, not the whole expression. In \(c \cdot b^0\) the zero sits on \(b\) alone, so \(c \cdot b^0 = c \times 1 = c\). But \((cb)^0 = 1\) because the parentheses hand the whole product to the zero. Likewise, \(-a^0 = -(a^0) = -1\) while \((-a)^0 = 1\). Before collapsing any zero power, check exactly what it is holding.
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 \(a\), $$a^{-n} = \frac{1}{a^n}.$$ A negative exponent moves the power to the denominator and makes the exponent positive. All three laws carry the negative sign without complaint. Multiplying \(a^m \times a^{-n}\) adds the exponents as usual, giving \(a^{m-n}\). Dividing by \(a^{-n}\) subtracts a negative, which adds, giving \(\dfrac{a^m}{a^{-n}} = a^{m+n}\). A fraction base with a negative exponent is the same as the reciprocal base with a positive one: $$\left(\frac{1}{a}\right)^n = a^{-n}.$$
Key idea Stacked exponents multiply, and the coefficient rides along. A tower of three or more exponents collapses by multiplying every exponent together: $$\left(\left(a^m\right)^n\right)^p = a^{m \times n \times p}.$$ An outer exponent also lands on any number riding in front of a variable, because the coefficient is one of the factors inside the parentheses: $$(c\,x^m)^n = c^n \, x^{mn}.$$ For example, \((2x^3)^4 = 16x^{12}\), not \(2x^{12}\). Keep the contrast straight. \((5^3)^2 = 5^6\) because stacking multiplies, while \(5^3 \times 5^2 = 5^5\) because multiplying separate powers of the same base adds.
Key idea Negative outer exponent flips a fraction. A fraction raised to a negative power turns over first, then the exponent becomes positive: $$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} = \frac{b^n}{a^n} \quad (a \ne 0,\, b \ne 0).$$ The negative sign is the reciprocal instruction. Once the fraction is upright, the positive exponent distributes over the new top and bottom as usual.
Key idea The common base strategy. When every base in an expression is really a power of the same number, rewrite them all in that one base first. Once a single base rules the whole expression, the three laws can fire without interference, and a tangle of different-looking numbers collapses to a single power. This is the master move of the chapter. Convert to a shared base, then let the laws handle the rest.
Every rule in Chapter 2 restates one idea: an exponent counts factors, and the laws describe how that count changes under each arithmetic operation. Multiplication concatenates 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 \(1\). A negative exponent is a count that crossed the fraction bar and went downstairs. That is the whole chapter in one picture.
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