If (a+1a )^2=3, then a^3+1a^3 equals:

a+1a=3 (a+1a)^3=33 a^3+3a^2a+3aa^2+1a^3 a^3+3(a+1a)+1a^3=33 a^3+1a^3+33=33 a^3+1a^3=0, 0

smc

algebra senior

Problem

(a + \frac{1}{a})^{2} = 3

, then

a^{3} + \frac{1}{a ^{3}}

equals:

(A)

\frac{10 3}{3}

(B)

33

(C)

0

(D)

77

Solution

a + \frac{1}{a} = 3 ⟹ (a + \frac{1}{a})^{3} = 33 ⟹ a^{3} + 3 \frac{a ^{2}}{a} + \frac{3 a}{a ^{2}} + \frac{1}{a ^{3}} ⟹ a^{3} + 3 (a + \frac{1}{a}) + \frac{1}{a ^{3}} = 33 ⟹ a^{3} + \frac{1}{a ^{3}} + 33 = 33 ⟹ a^{3} + \frac{1}{a ^{3}} = 0

0

Final answer