Simplify the following expression: 2/(1+a) + 1/(a-1) + (3a)/(1-a^2) + a/(1+a^3)
Answer (2)
To simplify the expression 1 + a 2 + a − 1 1 + 1 − a 2 3 a + 1 + a 3 a , we identify a common denominator to combine terms effectively. After factoring denominators and rewriting fractions, we find a single simplified expression. This process requires careful algebraic manipulation and attention to detail for accurate results.
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To simplify the expression given: 1 + a 2 + a − 1 1 + 1 − a 2 3 a + 1 + a 3 a , we'll approach it step-by-step and simplify each part.
Rewriting with Factored Denominators :
1 − a 2 3 a can be rewritten because 1 − a 2 is a difference of squares: 1 − a 2 = ( 1 − a ) ( 1 + a ) .
Similarly, 1 + a 3 a involves factoring the denominator. Sometimes such terms do not simplify easily without specific values for a , but assuming possible factoring is essential.
Combine Like Terms :
For fractions, finding a common denominator helps in combining them. Here it involves using the factored forms from step 1.
The common denominator generally is ( 1 + a ) ( a − 1 ) ( 1 − a 2 ) ( 1 + a 3 ) if simplification is difficult without specific values.
Simplifying :
Use any algebraic identities or simplifications wherever possible.
Recombine each part by considering equivalent expressions to see if they combine or simplify further.
Conclusion : The complexity of directly simplifying without further context or values given suggests evaluation or enjoyment of manipulating expressions with algebraic rules. This constitutes practice for algebra or rational expression handling in high school mathematics.
If you have any specific parts you find difficult or further numeric values to simplify the expression, feel free to share those for more concrete simplifications!
