c) If $x y z=1$, prove that $\frac{1}{1+x+y^{-1}}+\frac{1}{1+y+z^{-1}}+$\nd) If $a+b+c=0$, prove that $\frac{1}{1+x^a+x^b}+\frac{1}{1+x^b}+$\n12. a) If $x=2^{\frac{1}{3}}+2^{-\frac{1}{3}}$, prove that $2 x^3-6 x=5$.\nb) If $a=p^{\frac{1}{3}}-p^{-\frac{1}{3}}$, prove that $a^3+3 a=p-\frac{1}{p}$.\nc) If $x-2=3^{\frac{1}{3}}+3^{\frac{2}{3}}$, show that $x\left(x^2-6 x+3\right)=2$.\nOBJECTIVE QUESTION\n\nLet's tick $(\sqrt{ })$ the correct alternative.
Answer (2)
This response outlines how to prove algebraic identities involving sums and products using algebraic manipulation and substitution techniques. It systematically breaks down each proof step, showing how the relationships between variables can be used to simplify complex expressions. Understanding these techniques is crucial in advanced math and related fields.
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The solution involves proving several algebraic identities and equations using algebraic manipulation and given conditions.
Explanation
Introduction We are given several problems to prove. Let's address them one by one.
Examples
These types of algebraic proofs are fundamental in various fields such as physics, engineering, and computer science, where manipulating equations and simplifying expressions are essential skills. For instance, in physics, you might need to simplify complex equations to model the behavior of particles or waves. In engineering, you might use these techniques to optimize designs or analyze circuits. In computer science, these skills are crucial for algorithm design and analysis.
