Roots

Summary: Roots are inverse operations to powers, with the degree of the root matching the power to be undone; their behavior depends crucially on whether the degree is even or odd. (source: chapter-1.1.18, source: chapter-1.1.19)

Sources: chapter-1.1.11, chapter-1.1.12, chapter-1.1.15, chapter-1.1.18, chapter-1.1.19

Last updated: 2026-04-24

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General Idea

The $n$th root of a number is the quantity whose $n$th power reproduces that number. Euler treats square roots and cube roots as special cases of this general pattern. (source: chapter-1.1.18)

Even and Odd Degrees

For positive numbers, roots of all degrees are real. For negative numbers, even roots are imaginary while odd roots are real and negative. (source: chapter-1.1.18)

Rational and Irrational Cases

If the radicand is an exact power of the required degree, the root is rational. Otherwise the root is irrational. (source: chapter-1.1.12, source: chapter-1.1.15, source: chapter-1.1.18)

Exponent Form

Euler rewrites roots with fractional exponents: $\sqrt[n]{a}=a^{1/n}.$ (source: chapter-1.1.19)

This lets multiplication and division of roots follow the same laws as multiplication and division of powers. (source: chapter-1.1.19)

Related pages

• ch1.1.18-roots-and-powers
• powers-and-exponents
• irrational-numbers
• imaginary-numbers
• cube-roots