Question : If $(a+\frac{1}{a})^2=3$, the value of $(a^3+\frac{1}{a^3})$ is:
Option 1: $0$
Option 2: $3(a+\frac{1}{a})$
Option 3: $3(a^2+\frac{1}{a^2})$
Option 4: $1$
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Correct Answer: $0$
Solution :
Given: $(a+\frac{1}{a})^2=3$
The algebraic identity used is $(a+\frac{1}{a})^3=a^3+\frac{1}{a}^3+3(a+\frac{1}{a})$.
$(a+\frac{1}{a})^2=3$ (equation 1)
Take the square root of the given equation, we get,
$(a+\frac{1}{a})=\sqrt{3}$ (equation 2)
Multiplying equation 1 and equation 2, we get,
$(a+\frac{1}{a})^2(a+\frac{1}{a})=3\sqrt{3}$
⇒$(a+\frac{1}{a})^3=3\sqrt{3}$
⇒ $a^3+\frac{1}{a^3}+3(a+\frac{1}{a})=3\sqrt{3}$
⇒ $a^3+\frac{1}{a^3}+3\sqrt{3}=3\sqrt{3}$
⇒ $a^3+\frac{1}{a^3}=3\sqrt{3}–3\sqrt{3}$
⇒ $a^3+\frac{1}{a^3}=0$
Hence, the correct answer is 0.
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