Subtract.
$\frac{y}{10 a k^2}-\frac{5 a u^3}{4 b^2 k}$
Simplify your answer as much as possible.
Answer (1)
Find a common denominator for the two fractions, which is 20 a b 2 k 2 .
Rewrite each fraction with the common denominator: 20 a b 2 k 2 2 b 2 y and 20 a b 2 k 2 25 a 2 k u 3 .
Subtract the fractions: 20 a b 2 k 2 2 b 2 y − 25 a 2 k u 3 .
The simplified result is: 20 a b 2 k 2 2 b 2 y − 25 a 2 k u 3 .
Explanation
Understanding the Problem We are asked to subtract two fractions: 10 a k 2 y and 4 b 2 k 5 a u 3 . To do this, we need to find a common denominator.
Finding the Common Denominator The denominators are 10 a k 2 and 4 b 2 k . We need to find the least common multiple (LCM) of these two expressions.
Determining the LCD The prime factorization of 10 is 2 × 5 , and the prime factorization of 4 is 2 2 . Therefore, the LCM of 10 and 4 is 2 2 × 5 = 20 . The LCM of a and a is a . The LCM of k 2 and k is k 2 . The LCM of 1 and b 2 is b 2 . So, the least common denominator (LCD) is 20 a b 2 k 2 .
Rewriting Fractions with Common Denominator Now we rewrite each fraction with the LCD as the denominator: 10 a k 2 y = 10 a k 2 × ( 2 b 2 ) y × ( 2 b 2 ) = 20 a b 2 k 2 2 b 2 y 4 b 2 k 5 a u 3 = 4 b 2 k × ( 5 ak ) 5 a u 3 × ( 5 ak ) = 20 a b 2 k 2 25 a 2 k u 3
Subtracting the Fractions Now we subtract the two fractions: 20 a b 2 k 2 2 b 2 y − 20 a b 2 k 2 25 a 2 k u 3 = 20 a b 2 k 2 2 b 2 y − 25 a 2 k u 3
Simplifying the Result The resulting fraction is 20 a b 2 k 2 2 b 2 y − 25 a 2 k u 3 . We check if this fraction can be simplified further. There are no common factors in the numerator and denominator, so the fraction is already in its simplest form.
Final Answer Therefore, the final answer is: 20 a b 2 k 2 2 b 2 y − 25 a 2 k u 3
Examples
Fractions are used in everyday life, such as when calculating proportions in recipes, determining discounts while shopping, or understanding ratios in financial investments. This problem demonstrates how to subtract algebraic fractions, which is a fundamental skill in algebra and is used in more advanced mathematical concepts such as calculus and differential equations. For example, when dealing with rates of change or solving equations involving rational functions, proficiency in manipulating fractions is essential.
