Final answer:
The rational expression 10x÷20x-50x² is simplified by dividing the coefficients and subtracting the exponents of like terms, resulting in a simplified expression of ½ - ¼x.
Step-by-step explanation:
To simplify the given rational expression 10x÷20x-50x², first, we need to follow the correct order of operations. In this case, we need to divide the term in the numerator by the terms in the denominator before simplifying.
First, divide the constant term of the numerator (10) by the constant term of the denominator (20), which gives us 10/20 or ½. Next, deal with the variable x. When dividing like terms with exponents, subtract the exponents (keeping in mind that if there is no exponent written, it is implicitly 1). The x term in the numerator has an exponent of 1, and in the denominator, the x terms have exponents of 1 and 2, respectively. However, since we're only dividing by 20x initially, we subtract the exponents of x in 10x and 20x. This gives us x¹ - x¹ = x°, and since anything raised to the power of 0 equals 1, this term simplifies to just ½.
Now let's look at the second term in the denominator, -50x². To subtract this from the rational expression, you would typically need a common denominator, but we're dividing the 10x by the entire expression 20x-50x². Since division is distributive over subtraction, we actually divide 10x by each term separately: 10x/20x and 10x/(-50x²). After simplification, the second term becomes -⅔x or -¼, if we factor out x² from both terms in the denominator and simplify with 10x. Finally, we need to combine the results of the division. Since 10x ÷ 20x simplifies to ½ and 10x ÷ -50x² simplifies to -¼x, we get a simplified expression of ½ - ¼x when putting it all together. Eliminate terms wherever possible while simplifying. Check your answer to ensure it is reasonable.