[tex](2\sqrt5+3\sqrt7)^2=(2\sqrt5+3\sqrt7)(2\sqrt5+3\sqrt7)[/tex]
by definition of exponentiation;
[tex]=2\sqrt5(2\sqrt5+3\sqrt7)+3\sqrt7(2\sqrt5+3\sqrt7)[/tex]
[tex]=(2\sqrt5)^2+6\sqrt{35}+6\sqrt{35}+(3\sqrt7)^2[/tex]
by the distributive property of multiplication;
[tex]=2^2(\sqrt5)^2+3^2(\sqrt7)^2+12\sqrt{35}[/tex]
by the distributive property of exponentiation;
[tex]=20+63+12\sqrt{35}[/tex]
[tex]=83+12\sqrt{35}[/tex]
by grouping common terms and simplifying.
