SUMMARY
The expression \sqrt[3]{10+6\sqrt{3}}+\sqrt[3]{10-6\sqrt{3}} can be evaluated using the identity for the sum of cubes. By letting x = \sqrt[3]{10+6\sqrt{3}}+\sqrt[3]{10-6\sqrt{3}}, the equation simplifies to x^3 + 6x - 20 = 0. The Rational Root Theorem reveals that the only real solution is x = 2, confirming that the exact value of the expression is 2.
PREREQUISITES
- Understanding of cube roots and their properties
- Familiarity with the Rational Root Theorem
- Knowledge of binomial expansion, specifically
(a+b)^3
- Ability to manipulate and solve cubic equations
NEXT STEPS
- Study the Rational Root Theorem and its applications in polynomial equations
- Learn about binomial expansion and its relevance in algebraic simplifications
- Practice solving cubic equations using various methods, including factoring
- Explore advanced topics in algebra such as surds and their simplifications
USEFUL FOR
Students, educators, and anyone interested in algebraic problem-solving, particularly those dealing with cubic equations and radical expressions.
