Simplify.
$\frac{5}{x^2(x-2)}+\frac{x}{x-2}$
$\frac{x^{[?]}+\square}{x^2(x-\square)}$
Answer (1)
Find a common denominator: x 2 ( x − 2 ) .
Rewrite the second fraction with the common denominator: x − 2 x = x 2 ( x − 2 ) x 3 .
Add the fractions: x 2 ( x − 2 ) 5 + x 2 ( x − 2 ) x 3 = x 2 ( x − 2 ) x 3 + 5 .
The simplified expression is x 2 ( x − 2 ) x 3 + 5 .
Explanation
Understanding the Problem We are given the expression x 2 ( x − 2 ) 5 + x − 2 x and we want to simplify it and express it in the form x 2 ( x − □ ) x [ ?] + □ .
Finding the Common Denominator To add the two fractions, we need a common denominator. The common denominator is x 2 ( x − 2 ) .
Rewriting the Second Fraction We rewrite the second fraction with the common denominator: x − 2 x = x 2 ( x − 2 ) x ⋅ x 2 = x 2 ( x − 2 ) x 3 .
Adding the Fractions Now we add the two fractions: x 2 ( x − 2 ) 5 + x 2 ( x − 2 ) x 3 = x 2 ( x − 2 ) 5 + x 3 = x 2 ( x − 2 ) x 3 + 5 .
Identifying the Missing Values Comparing this to the desired form x 2 ( x − □ ) x [ ?] + □ , we can identify the missing values. The exponent of x in the numerator is 3, the constant term in the numerator is 5, and the constant term in the denominator is 2.
Final Answer Therefore, the simplified expression is x 2 ( x − 2 ) x 3 + 5 .
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used in many areas of mathematics and engineering. For example, when analyzing electrical circuits, you often encounter complex fractions involving impedances. Simplifying these fractions allows you to determine the overall impedance of the circuit and analyze its behavior. Similarly, in physics, when dealing with equations involving forces and motion, simplifying rational expressions can help in solving for unknown variables and understanding the relationships between different physical quantities.
