Cube Roots

Summary: Cube roots are third roots, remain real for negative inputs, and form a distinct family of irrational quantities when the radicand is not a perfect cube. (source: chapter-1.1.14, source: chapter-1.1.15, source: chapter-1.1.18)

Sources: chapter-1.1.14, chapter-1.1.15, chapter-1.1.18, chapter-1.2.9

Last updated: 2026-04-28

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Definition

The cube root of a number is the quantity whose cube equals that number. (source: chapter-1.1.15)

Exact and Inexact Cases

Perfect cubes such as $8$, $27$, and $-125$ have exact cube roots $2$, $3$, and $-5$. When the radicand is not a perfect cube, the result is irrational. (source: chapter-1.1.15)

Operations

Euler treats cube roots with the same product and quotient logic used for square roots: $\sqrt[3]{a}\sqrt[3]{b}=\sqrt[3]{ab},\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}.$ (source: chapter-1.1.15)

He also rewrites extracted cube factors outside the radical, as in $\sqrt[3]{16}=2\sqrt[3]{2}$. (source: chapter-1.1.15)

Sign Rule

Unlike square roots, cube roots of negative numbers are real: $\sqrt[3]{-a}=-\sqrt[3]{a}.$ (source: chapter-1.1.15, source: chapter-1.1.18)

Cube Root Extraction of Compound Quantities

Chapter 1.2.9 gives a systematic algorithm for extracting the cube root of a polynomial (§335–337): (source: chapter-1.2.9)

1. Take the cube root of the leading term as the first term $a$ of the root.
2. Subtract $a^3$; the remainder equals $(3a^2+3ab+b^2)b$.
3. Divide by the trial divisor $3a^2$ to obtain the next term $b$.
4. Complete the divisor to $3a^2+3ab+b^2$ and verify zero remainder.

The same algorithm underlies numerical cube root extraction. Examples: $\sqrt[3]{2197}=13$ and $\sqrt[3]{34965783}=327$. (source: chapter-1.2.9)

Related pages

• ch1.1.14-cubic-numbers
• ch1.1.15-cube-roots
• ch1.2.9-cubes-and-cube-root-extraction
• roots
• irrational-numbers
• imaginary-numbers
• powers-and-exponents