ch1.1.17-calculation-of-powers

Chapter XVII – Of the Calculation of Powers

Summary: Derives the laws for multiplying, dividing, and re-raising powers by operating directly on exponents. (source: chapter-1.1.17)

Sources: chapter-1.1.17

Last updated: 2026-04-24

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§180: Addition and Subtraction

Different powers generally remain formal sums or differences, such as $a^3+a^2$ or $a^5-a^4$. Only like powers combine, for example: $a^3+a^3=2a^3.$ (source: chapter-1.1.17)

§181–186: Multiplication and Division of Powers

Multiplying a power by $a$ increases its exponent by $1$, multiplying by $a^2$ increases it by $2$, and so on. In general: $a^m\cdot a^n=a^{m+n}.$ (source: chapter-1.1.17)

Euler applies the same rule to negative exponents, so $a^{-1}\cdot a=a^0=1$. (source: chapter-1.1.17)

Division subtracts exponents: $\frac{a^m}{a^n}=a^{m-n}.$ (source: chapter-1.1.17)

Thus $\frac{a^6}{a^7}=a^{-1}$ and $\frac{a^{-3}}{a^4}=a^{-7}$. (source: chapter-1.1.17)

§187–188: Powers of Powers

Raising a power to another power multiplies exponents: ${\left(a^m\right)}^n=a^{mn}.$ (source: chapter-1.1.17)

Euler illustrates this with: ${\left(a^3\right)}^2=a^6,{\left(a^4\right)}^3=a^{12}.$ (source: chapter-1.1.17)

He also notes older names such as biquadrate for the fourth power, square-cubed for the sixth, and cubo-cube for the ninth. (source: chapter-1.1.17)

Related pages

• powers-and-exponents
• roots
• logarithms
• ch1.1.16-powers-in-general