Subtract. [tex]$\frac{v}{3 c k^3}-\frac{4 c^3 x^3}{9 a^3 k^2}$[/tex]
Simplify your answer as much as possible.
Answer (1)
Find a common denominator for the two fractions, which is 9 a 3 c k 3 .
Rewrite each fraction with the common denominator: 3 c k 3 v = 9 a 3 c k 3 3 a 3 v and 9 a 3 k 2 4 c 3 x 3 = 9 a 3 c k 3 4 c 4 k x 3 .
Subtract the fractions: 9 a 3 c k 3 3 a 3 v − 9 a 3 c k 3 4 c 4 k x 3 = 9 a 3 c k 3 3 a 3 v − 4 c 4 k x 3 .
The simplified result is: 9 a 3 c k 3 3 a 3 v − 4 c 4 k x 3 .
Explanation
Problem Analysis We are asked to subtract two fractions: 3 c k 3 v and 9 a 3 k 2 4 c 3 x 3 . To do this, we need to find a common denominator and combine the fractions.
Finding the Common Denominator The denominators are 3 c k 3 and 9 a 3 k 2 . The least common multiple (LCM) of these denominators is 9 a 3 c k 3 .
Rewriting the First Fraction We rewrite each fraction with the common denominator 9 a 3 c k 3 . The first fraction becomes 3 c k 3 v = 3 c k 3 × 3 a 3 v × 3 a 3 = 9 a 3 c k 3 3 a 3 v .
Rewriting the Second Fraction The second fraction becomes 9 a 3 k 2 4 c 3 x 3 = 9 a 3 k 2 × c k 4 c 3 x 3 × c k = 9 a 3 c k 3 4 c 4 k x 3 .
Subtracting the Fractions Now, we subtract the two fractions: 9 a 3 c k 3 3 a 3 v − 9 a 3 c k 3 4 c 4 k x 3 = 9 a 3 c k 3 3 a 3 v − 4 c 4 k x 3 .
Final Result We check if the numerator and denominator have any common factors to simplify the fraction further. In this case, there are no common factors. Therefore, the simplified result is: 9 a 3 c k 3 3 a 3 v − 4 c 4 k x 3 .
Examples
When calculating the total resistance in parallel circuits, you often encounter expressions involving fractions. Subtracting such fractions, as we did here, helps simplify the expression and find the equivalent resistance. This skill is also useful in various engineering and physics problems where you need to combine or compare quantities expressed as fractions with different denominators. For example, in fluid dynamics, you might need to subtract flow rates or pressures given as rational expressions.
