Question

Suppose a is a real number for which (a^2)+(1/a^2)=14. What is the largest possible value of (a^3)+(1/a^3)?

Answer 1

Step-by-step explanation:

${a}^{2} + ({ \frac{1}{a} })^{2} = 14 \\ since \: \\ (a \: + \frac{1}{a})^{2} = {a}^{2} + ( { \frac{1}{a} )}^{2} + 2 \\ = 14 + 2 = 16 \\ so \\ a \: + \frac{1}{a} = \sqrt{16} = 4 \\ therefore \\ {a}^{3} + ({ \frac{1}{a} })^{3} = {(a + \frac{1}{a} })^{3} - 3(a + \frac{1}{a} ) \\ = {4}^{3} - 12 = 64 - 12 = 52$