Overview
A monomial is a product of a coefficient and variable powers. Multiply coefficients and add exponents of equal bases. In division, cancel common factors and subtract exponents, preserving the original denominator restrictions.
Consider this example: Simplify (−3a²b)(4ab³). The coefficients multiply to −12. The a powers add to 3 and the b powers add to 4, giving −12a³b⁴. Adding coefficients would produce 1 instead of −12.
The method also handles a different case: For x ≠ 0, simplify 18x⁵/(−6x²). The coefficient quotient is −3, and cancellation leaves x³. Therefore the expression is −3x³ for x ≠ 0. The original denominator still excludes 0.
For a further application, consider: For nonzero x, the expression (kx²)(3x³)/(6x⁴) equals 5x for every x. What is k? The variable powers simplify to x^(2+3−4) = x. The coefficient is k/2, so k/2 = 5 and k = 10. Matching 5 directly to k overlooks the coefficient 3/6.
Adding coefficients in a product or multiplying variable exponents confuses different operations. Cancellation can simplify a formula without making an originally excluded input valid. Separate coefficient arithmetic from variable-factor arithmetic. Check a new quotient’s domain before cancellation and use small nonzero inputs to compare original and simplified forms.