Final answer:
To simplify a complex fraction, find a common denominator, combine like terms, apply the properties of exponents, and reduce the expression to its simplest form.
Step-by-step explanation:
To simplify the complex fraction 5 - 2/x - 4 divided by 5/x - 4 + 2, we must follow the correct mathematical steps. Notice how the units of the numerator in one fraction could potentially cancel with the units of the denominator in the other, but in this case, we are dealing mainly with the simplification and addition/subtraction of polynomials and rational expressions.
First, identify the common denominator for both the numerator and the denominator expression, which in this case is x - 4. Convert both expressions to have this common denominator, then simplify the resulting expression by combining like terms and reducing the complex fraction to a simpler form. This might involve multiplying both the numerator and the denominator by the LCD to eliminate the fractions within fractions present in the complex fraction.
For example, if you had a complex fraction like (a/b) / (c/d), you would multiply both the numerator and the denominator by bd to get rid of the fraction within a fraction. The same principle applies here.
Once simplified, if there are any exponential terms, we can simplify further by applying the division of exponentials rule, which says to subtract the exponents of like bases when dividing. For example, in the expression 1 x^n, a negative exponent flips the base to the denominator, converting multiplication to division.
In summary, we carefully combine similar terms, apply the properties of exponents when needed, and always ensure to verify our steps for correct cancellation and simplification to reach the final simplified form.