If \(a+\displaystyle\frac{1}{a}=2\), then find the value of \(2a^{40}+\displaystyle\frac{3}{a^{15}}\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 5
Clearing the fraction turns the condition into a perfect square equal to zero, so \(a\) has only one value, \(a=1\). Every power of 1 is 1.
Multiplying by \(a\): \(a^2+1=2a\), so \(a^2-2a+1=0\)
\((a-1)^2=0\), so \(a=1\)
\(a^{40}=1\) and \(a^{15}=1\), so \(2a^{40}=2\) and \(\displaystyle\frac{3}{a^{15}}=3\)
\(2a^{40}+\displaystyle\frac{3}{a^{15}}=2+3=5\)