Exponent Rules, and Why They Work

x^{a} \cdot x^{b} = x^{a + b}

\frac{x ^{a}}{x ^{b}} = x^{a - b}

(x^{a})^{b} = x^{ab}

x^{- a} = \frac{1}{x ^{a}}

You can see the first rule by writing it out: $x^{2} \cdot x^{3}$ is $(x \cdot x) (x \cdot x \cdot x)$ , which has five factors of $x$ .

Worked examples

$x^{3} \cdot x^{4} = x^{7}$

$\frac{x ^{6}}{x ^{2}} = x^{4}$

$(x^{2})^{5} = x^{10}$

$(2 x^{3})^{2} = 2^{2} \cdot x^{6} = 4 x^{6}$

$x^{2 - 5} = x^{- 3}$

$x^{- 3} = \frac{1}{x ^{3}}$

$(x + y)^{2} = x^{2} + y^{2}$

Exponents don't distribute over addition. Multiply it out: $(x + y) (x + y) = x^{2} + 2 x y + y^{2}$ .

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