AI Tutor
If 232^3 means multiplying three twos together to get 88, how do you multiply a number by itself negative three times?
You don't — exponents are not just repeated multiplication, but instructions on a number line of repeated operations.

The Shrinking Pattern

Look at what happens when powers of 22 step downward: 23=82^3 = 8, 22=42^2 = 4, and 21=22^1 = 2. Each step down divides the value by 22.
Following that exact division pattern past zero gives 20=12^0 = 1, so 2−1=122^{-1} = \frac{1}{2}, and 2−2=142^{-2} = \frac{1}{4}.
In 1655, English mathematician John Wallis formalized this pattern so that multiplication and division rules would hold true across all numbers.
A reciprocal is simply the multiplicative inverse of a number, meaning ab\frac{a}{b} flipped to ba\frac{b}{a}. A negative exponent tells you to take the reciprocal of the base raised to the positive power: x−n=1xnx^{-n} = \frac{1}{x^n}.
What happens when a coefficient sits right in front of that negative exponent?

Watching the Base

A coefficient is the numerical multiplier placed in front of a variable term, like the 55 in 5x−35x^{-3}.
Exponents apply only to their immediate base unless parentheses say otherwise, so in 5x−35x^{-3}, only xx has the exponent −3-3.