ch1.1.16-powers-in-general

Chapter XVI – Of Powers in general

Summary: Generalizes squares and cubes into the notion of powers, introduces exponent notation $a^n$, and explains zero, negative, product, fraction, and negative-base exponents. (source: chapter-1.1.16)

Sources: chapter-1.1.16

Last updated: 2026-04-24

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§168–174: General Idea of Powers and Exponents

A power is formed by multiplying a number by itself one or more times. Squares are second powers and cubes are third powers. (source: chapter-1.1.16)

Euler then adopts exponent notation: $a^2,a^3,a^4,\dots ,a^{100}.$ (source: chapter-1.1.16)

The raised number is the exponent, and $a^1$ is simply $a$. Euler also describes the sequence $a,a^2,a^3,\dots$ as a geometrical progression. (source: chapter-1.1.16)

§175–177: Zero and Negative Exponents

Working backward in the sequence of powers leads Euler to the rule: $a^0=1.$ (source: chapter-1.1.16)

Continuing further backward gives negative exponents: $a^{-1}=\frac{1}{a},a^{-2}=\frac{1}{a^2},a^{-3}=\frac{1}{a^3}.$ (source: chapter-1.1.16)

§178–179: Products, Fractions, and Negative Bases

The powers of a product and of a fraction are formed factorwise: ${\left(ab\right)}^n=a^n{b}^n,{\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}.$ (source: chapter-1.1.16)

For a negative base $-a$, the signs alternate: $(-a{)}^1=-a,(-a{)}^2=+a^2,(-a{)}^3=-a^3,(-a{)}^4=+a^4.$ (source: chapter-1.1.16)

Thus odd powers of a negative number are negative, while even powers are positive. (source: chapter-1.1.16)

Related pages

• powers-and-exponents
• roots
• positive-negative-numbers
• logarithms