Exponents

An exponent counts repeated multiplication: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$. The exponent rules all follow from counting factors. Multiplying $a^3 \cdot a^4$ strings together seven factors of $a$, so the exponents add.

Zero and negative exponents extend the same pattern. Dividing $a^3$ by $a^3$ gives 1, and by the quotient rule it is also $a^0$, so $a^0 = 1$ for $a \ne 0$. Dividing once more by $a$ gives $a^{-1} = \frac{1}{a}$. Negative exponents mean reciprocals, not negative numbers.

Watch what the exponent is attached to. In $-3^2$ the exponent applies only to the 3, so the value is $-9$; in $(-3)^2$ it applies to the whole bracket, giving $9$. Exponents also make very large and very small quantities manageable: the distance to the Sun, about $1.5 \times 10^{11}$ metres, and the width of a bacterium, about $2 \times 10^{-6}$ metres, are both easy to compare in scientific notation.