Final answer:
To rationalize the denominator of 1/(4-√7)², we first expand the perfect square to get 23 - 8√7. The expression becomes 1 / (23 - 8√7), which has a rationalized denominator.
Step-by-step explanation:
To rationalize the denominator and simplify the expression 1/(4-√7)², we need to eliminate the square root from the denominator. Considering the given expression is a perfect square, we realize that squaring a binomial expands according to the formula (a - b)² = a² - 2ab + b². Here, a is 4 and b is √7.
Expanding the expression, (4 - √7)² becomes 4² - 2(4)(√7) + (√7)² which simplifies to 16 - 8√7 + 7. Simplifying further, the expression becomes 23 - 8√7. Now the fraction is 1 / (23 - 8√7).
Since eliminate terms wherever possible to simplify the algebra, we see that no terms cancel out in this case, but we have a rationalized denominator. To check if the result is reasonable, we confirm that there are no square roots in the denominator and the expression is indeed simplified.