Reduce the coefficients to lowest terms. The coefficients are the numbers written on the variables' left-hand sides. For instance, in [4(x^2)(y^3)(z)] / [6(x)(y^5)(z)], the coefficients are 4 and 6. To reduce 4/6 to lowest terms, divide both the numerator and denominator by their greatest common factor, which is the largest integer that divides into both. The greatest common factor of 4 and 6 is 2, and dividing both parts of the fraction by 2 produces a result of 2/3. Thus, [4(x^2)(y^3)(z)] / [6(x)(y^5)(z)] becomes [2(x^2)(y^3)(z)] / [3(x)(y^5)(z)].
Cancel out any like variables whose bases and exponents match exactly. In [2(x^2)(y^3)(z)] / [3(x)(y^5)(z)], a "z" appears in both the numerator and denominator, and its exponent in both parts of the fraction is just 1. Hence, cross out each "z," rendering [2(x^2)(y^3)] / [3(x)(y^5)]. Do not cancel the "x" or "y" terms because their exponents aren't identical.
Identify powers with the same base and divide them by subtracting their exponents. In the example, "x^2" and "x" share a common base, "x," and "y^3" and "y^5" share a common base, "y." Subtract the exponent of the variable in the denominator from that of the variable in the numerator. With the "x" terms, subtract 2 from 1, yielding x^1, which simplifies to just x. With the "y" terms, subtract 3 from 5, yielding y^-2. Write terms with positive exponents, such as "x" in this example, in the numerator of the fraction. Write terms with negative exponents in the fraction's denominator and change the exponent to a positive. In the example, y^-2 becomes y^2 when written in the denominator. Hence, the example becomes (2x) / (3y^2).
Ensure that no shared variables exist between the numerator and denominator. If they do, cancel them out as described in Step 2, or divide them by subtracting their exponents as described in Step 3. Also ensure that all exponents are positive. In the example, there are no shared variables and all exponents are positive, so (2x) / (3y^2) is the final answer.