(a⁻¹ + b⁻¹)² - 4a⁻¹b⁻¹

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To simplify the expression (a⁻¹ + b⁻¹)² - 4a⁻¹b⁻¹, we first need to expand the squared term using the formula (a + b)² = a² + 2ab + b²:

(a⁻¹ + b⁻¹)² = (a⁻¹)² + 2(a⁻¹)(b⁻¹) + (b⁻¹)²
= a⁻² + 2(a⁻¹b⁻¹) + b⁻²

Now, we can rewrite the expression using this expansion:

(a⁻² + 2(a⁻¹b⁻¹) + b⁻²) - 4(a⁻¹b⁻¹)

Next, we simplify the expression:

a⁻² + 2(a⁻¹b⁻¹) + b⁻² - 4(a⁻¹b⁻¹)
= a⁻² + 2a⁻¹b⁻¹ + b⁻² - 4a⁻¹b⁻¹
= a⁻² + b⁻² - 2a⁻¹b⁻¹

Therefore, the simplified expression is a⁻² + b⁻² - 2a⁻¹b⁻¹.