Towers of powers and negative exponents

Two situations lend themselves easily to mistakes: “tower” powers (an exponent of an exponent) and powers with a negative exponent of a fraction.

Beware

• $3^{2^3} = 3^8 = 6561$, not $(3^2)^3 = 9^3 = 729$. Without brackets, the power written higher up is carried out first!
• $\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$: a negative exponent on a fraction is equivalent to swapping numerator and denominator and changing the sign of the exponent.

In the first case the tower $a^{b^c}$ is read from the top: first $b^c$ is calculated, then $a$ is raised to that result.

Links

Topics: Numbers and operations
Concepts: Power · Properties of powers
Skills: Calculating