Powers and Exponents

Summary: Powers generalize repeated multiplication, and exponents provide the notation and laws that unify ordinary powers, reciprocals, and roots. (source: chapter-1.1.16, source: chapter-1.1.17, source: chapter-1.1.19)

Sources: chapter-1.1.16, chapter-1.1.17, chapter-1.1.19, chapter-1.1.21

Last updated: 2026-04-24

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Basic Notation

Euler writes powers as $a^2$, $a^3$, $a^4$, and in general $a^n$, where $n$ is the exponent. (source: chapter-1.1.16)

Foundational Laws

The central rules are: $a^m{a}^n=a^{m+n},\frac{a^m}{a^n}=a^{m-n},{\left(a^m\right)}^n=a^{mn}.$ (source: chapter-1.1.17)

Euler also makes explicit that: $a^0=1,a^{-n}=\frac{1}{a^n}.$ (source: chapter-1.1.16)

Fractional Exponents

Roots are written with fractional exponents: $a^{1/2}=\sqrt{a},a^{1/3}=\sqrt[3]{a},a^{m/n}=\sqrt[n]{a^m}.$ (source: chapter-1.1.19)

This makes radicals part of the same exponent system rather than a separate notation. (source: chapter-1.1.19)

Connection to Logarithms

Euler later defines logarithms precisely by asking for the unknown exponent in an expression of the form $a^b=c$. (source: chapter-1.1.20, source: chapter-1.1.21)

Related pages

• ch1.1.17-calculation-of-powers
• ch1.1.20-methods-of-calculation
• roots
• irrational-numbers
• logarithms
• square-roots-and-irrational-numbers