Simplify (2√5 + 3√7)^2. Show all your work. Justify each step.
To simplify (2√5 + 3√7)^2, we will use the distributive property to expand the square of the binomial:
(2√5 + 3√7)^2
= (2√5 + 3√7)(2√5 + 3√7) [using the definition of squaring]
= 4(√5)^2 + 6√5√7 + 6√5√7 + 9(√7)^2 [expanding using the distributive property]
= 4(5) + 12√35 + 9(7) [simplifying the square roots]
= 20 + 12√35 + 63 [calculating the products]
= 83 + 12√35 [combining like terms]
Therefore, (2√5 + 3√7)^2 simplifies to 83 + 12√35.