Errors with Negative Exponents: Incorrectly interpreting or manipulating negative exponents, such as \(a^{-n} \neq -a^n\) or misapplying rules when factoring.
Incorrectly Identifying the GCF with Rational/Negative Exponents: When factoring expressions like \(3(1+x)^{1/3} - x(1+x)^{-2/3}\), choosing the wrong power for the GCF \((1+x)\). The rule is to factor out the term with the smallest (most negative or least positive) exponent.
Errors in Subtracting Exponents During Factoring: When factoring out a GCF like \(A^m\) from \(A^M\), the remaining exponent is \(M-m\). Mistakes often occur with signs, especially if \(m\) is negative (e.g., \(1/3 - (-2/3) = 1/3 + 2/3\)).
Distributing Unnecessarily in Denominators: When simplifying complex fractions or expressions resulting from the quotient rule, distributing terms in the overall denominator can make further simplification or cancellation harder. It's often better to keep denominators factored.
Algebraic Errors in Simplifying Compound Fractions: When multiplying by LCD to clear minor denominators, making errors in distribution or combining terms.
Not Factoring Completely: After an initial factoring step, failing to see if any resulting polynomial factors (like a quadratic) can be factored further.