Simplifying Complex Expressions: (b-c)/(a^2-bc^2) (ca)/(b^2-ca^2) (ab)/(ab^2-c^2)
One common challenge in algebra is simplifying complex expressions. In this article, we'll guide you through the process of simplifying the following expression:
Simplification Process
The given expression is:
(b-c)/(a^2-bc^2) (ca)/(b^2-ca^2) (ab)/(ab^2-c^2)
Step 1: Factoring in the Denominators
First, let's factor the denominators of each term of the expression.
Factoring the First Term
The expression is: (b-c)/(a^2-bc^2)
Note that a^2-bc^2 can be factored using the difference of squares formula:
a^2 - bc^2 (a - u03bc) * (a u03bc)
Factoring the Second Term
The expression is: (ca)/(b^2-ca^2)
Note that b^2-ca^2 can be factored as follows:
b^2 - ca^2 (b - u03bd)(b u03bd)
Factoring the Third Term
The expression is: (ab)/(ab^2-c^2)
Note that ab^2-c^2 can be factored as follows:
ab^2 - c^2 (ab - u03be)(ab u03be)
Step 2: Simplifying Each Term
Now that we have factored the denominators, let's simplify each term of the expression.
Simplifying the First Term
math frac{b-c}{a^2 - bc^2} frac{b-c}{(a - boldsymbol{c})(a boldsymbol{c})} lt;/math
Simplifying the Second Term
math frac{ca}{b^2 - ca^2} frac{ca}{(b - boldsymbol{a})(b boldsymbol{a})} lt;/math
Simplifying the Third Term
math frac{ab}{ab^2 - c^2} frac{ab}{(ab - boldsymbol{c})(ab boldsymbol{c})} lt;/math
Step 3: Combining the Terms
Now, we combine all the simplified terms together:
math 4bc(b - c)/a(b - c) a(b - c)/b(b - c) a(b - c)/a(b - c) 4b c b - c/a (b - c) a - b c a^2 - b^2 - 2b c - c^2 /math
Final Simplified Expression
The final simplified expression is: 4 b c (b - c)/a (b - c) a (b - c)/b (b - c) a (b - c)/a (b - c) 4b c b - c/a (b - c) a - b c a^2 - b^2 - 2b c - c^2
Conclusion
By following these steps, we can simplify complex algebraic expressions. Understanding and practicing these steps will improve your algebra skills and help you tackle similar problems in mathematics and other related fields.