AI Tutor
If x3x^3 means multiplying three factors of xx, what could it possibly mean to multiply by xx negative three times? It sounds like nonsense until you realize that exponents describe an operation direction, not just counting objects.
A negative exponent indicates the multiplicative inverse, or reciprocal, of the base raised to the positive power: for any nonzero base xx, x−n=1xnx^{-n} = \frac{1}{x^n}.
We can prove this directly with the quotient rule of exponents, which states that xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}. Notice what happens when a<ba < b:
English mathematician John Wallis first introduced this notation in 1655 in his work Arithmetica Infinitorum, establishing that negative indices smoothly extend arithmetic patterns below zero.
So a negative power in the numerator sends factors to the denominator — but what happens when a negative exponent is already sitting downstairs in a fraction?

Crossing the Fraction Bar

When a negative power appears in the denominator, taking its reciprocal moves that base up to the numerator: 1x−n=1÷1xn=1⋅xn1=xn\frac{1}{x^{-n}} = 1 \div \frac{1}{x^n} = 1 \cdot \frac{x^n}{1} = x^n.
Every variable factor with a negative exponent can simply change levels across the fraction bar, converting its exponent into a positive number.
Now that we know how single terms relocate, how do we handle expressions with multiple variables and numerical coefficients?

Simplifying Multi-Variable Expressions

Let's simplify the algebraic fraction 12x−4y318x2y−5\frac{12x^{-4}y^3}{18x^2y^{-5}} so that all exponents are strictly positive.