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Simplify(2√5+3√7)^2
Show your work. Justify each step.
since (a+b)^2 = a^2 + 2ab + b^2, we have
(2√5+3√7)^2
= (2√5)^2 + 2(2√5)(3√7) + (3√7)^2
= 4*5 + 12√5√7 + 9*7
= ...
To simplify the expression (2√5 + 3√7)^2, we can use the distributive property of exponents.
Step 1: Square each term inside the parentheses separately.
(2√5)^2 + 2*(2√5)*(3√7) + (3√7)^2
Step 2: Simplify the squares.
4 * 5 + 2*(2*3*√5√7) + 9 * 7
Step 3: Simplify the square roots.
20 + 12√35 + 63
Step 4: Combine like terms.
83 + 12√35
So, the simplified form of (2√5 + 3√7)^2 is 83 + 12√35.
Please note that in Step 2, we used the property √a * √b = √(a * b) to simplify the expression (2√5)*(3√7) to 2*(2*3)*√5√7. In Step 3, we simplified the product to 12√35.