Question

Simplify (2sqrt5+3sqrt7)^2. Show your work. Justify each step

Answer 1

$(2\sqrt5+3\sqrt7)^2=(2\sqrt5+3\sqrt7)(2\sqrt5+3\sqrt7)$

by definition of exponentiation;

$=2\sqrt5(2\sqrt5+3\sqrt7)+3\sqrt7(2\sqrt5+3\sqrt7)$
$=(2\sqrt5)^2+6\sqrt{35}+6\sqrt{35}+(3\sqrt7)^2$

by the distributive property of multiplication;

$=2^2(\sqrt5)^2+3^2(\sqrt7)^2+12\sqrt{35}$

by the distributive property of exponentiation;

$=20+63+12\sqrt{35}$
$=83+12\sqrt{35}$

by grouping common terms and simplifying.