If y + $\frac{1}{y}$ = 9, evaluate y2 + $\frac{1}{y^2}$

Analyze the question: We are given y + 1/y = 9 and asked to find y² + 1/y².

Concept: Square the expression y + 1/y. This produces the y² + 1/y² term plus a constant:

(y + 1/y)² = y² + 2(y)(1/y) + 1/y² = y² + 2 + 1/y²

Solution:

1. Square both sides of the given equation: (y + 1/y)² = 9² = 81

2. Expand the left side: y² + 2 + 1/y² = 81

3. Subtract 2 from both sides: y² + 1/y² = 81 − 2 = 79

Answer: E (79)

Tip: Whenever you see y + 1/y = k, the answer to y² + 1/y² is always k² − 2. Likewise, if you see y − 1/y = k, then y² + 1/y² = k² + 2. Many students wrongly say 9² = 81 and stop, or forget to subtract the 2. Always remember the middle term 2·y·(1/y) = 2.