ch1.1.19-fractional-exponents

Chapter XIX – Of the Method of representing Irrational Numbers by Fractional Exponents

Summary: Rewrites radicals using fractional exponents, explains how to interpret $a^{m/n}$, and uses exponent rules to multiply, divide, and simplify surds. (source: chapter-1.1.19)

Sources: chapter-1.1.19

Last updated: 2026-04-24

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§195–200: Roots as Fractional Powers

Euler derives: $a^{1/2}=\sqrt{a},a^{1/3}=\sqrt[3]{a},a^{1/4}=\sqrt[4]{a}.$ (source: chapter-1.1.19)

More generally, an $n$th root may be written as $a^{1/n}$, and this notation is presented as a natural continuation of the usual laws of exponents. (source: chapter-1.1.19)

§201–203: Interpreting General Fractional Exponents

An exponent such as $a^{4/3}$ means: take the fourth power and then the cube root, or equivalently combine the two operations in the order justified by exponent laws. (source: chapter-1.1.19)

Examples include: $a^{4/3}=\sqrt[3]{a^4},a^{3/4}=\sqrt[4]{a^3},a^{5/2}=a^2\sqrt{a}.$ (source: chapter-1.1.19)

Negative fractional exponents denote reciprocals of radicals: $a^{-1/2}=\frac{1}{\sqrt{a}},a^{-1/3}=\frac{1}{\sqrt[3]{a}}.$ (source: chapter-1.1.19)

§204–205: Common Indices and Calculations with Surds

Euler notes that a root can be written in many equivalent radical forms: $\sqrt{a}=\sqrt[4]{a^2}=\sqrt[6]{a^3}.$ (source: chapter-1.1.19)

This allows surds with different indices to be reduced to a common index before multiplication or division. The same results also follow directly from adding or subtracting fractional exponents. (source: chapter-1.1.19)

Related pages

• powers-and-exponents
• roots
• irrational-numbers
• square-roots-and-irrational-numbers