Answer
a) $3\sqrt2$
b) $2\sqrt7$
c) $\sqrt {(2\sqrt {p^2}+2\sqrt {q^2})}$
Work Step by Step
a)
$(\sqrt 7+\sqrt 2)^2+ (\sqrt 7-\sqrt 2)^2=x^2$
$(\sqrt 7*\sqrt7+\sqrt7*\sqrt 2+\sqrt 2*\sqrt7+\sqrt2*\sqrt 2)+(\sqrt 7*\sqrt7-\sqrt7*\sqrt 2-\sqrt 2*\sqrt7+\sqrt2*\sqrt 2)$
$(\sqrt {49}+\sqrt{14}+\sqrt {14}+\sqrt4+\sqrt {49}-\sqrt{14}-\sqrt {14}+\sqrt4)$
$(\sqrt {49}+\sqrt4+\sqrt {49}+\sqrt4)$
$7+2+7+2=18$
$x^2=18$
$\sqrt {x^2} = sqrt{18}$
$\sqrt{18} = \sqrt{3*3*2}$
$\sqrt {3*3*2} = 3\sqrt2$
b)
$(\sqrt {11}+\sqrt 3)^2+ (\sqrt {11}-\sqrt 3)^2=x^2$
$(\sqrt {11}*\sqrt{11}+\sqrt{11}*\sqrt 3+\sqrt 3*\sqrt{11}+\sqrt3*\sqrt 3)+(\sqrt {11}*\sqrt{11}-\sqrt{11}*\sqrt 3-\sqrt 3*\sqrt{11}+\sqrt3*\sqrt 3)$
$(\sqrt {121}+\sqrt{33}+\sqrt {33}+\sqrt9+\sqrt {121}-\sqrt{33}-\sqrt {33}+\sqrt9)$
$11+3+11+3 = 28$
$x^2=28$
$\sqrt {x^2} = \sqrt{28}$
$x= \sqrt{28}$
$x= \sqrt{2*2*7}$
$x=2*\sqrt7$
c)
$(\sqrt {p}+\sqrt q)^2+ (\sqrt {p}-\sqrt q)^2=x^2$
$(\sqrt {p}*\sqrt{p}+\sqrt{p}*\sqrt q+\sqrt q*\sqrt{p}+\sqrt q*\sqrt q)+(\sqrt {p}*\sqrt{p}-\sqrt{p}*\sqrt q-\sqrt q*\sqrt{p}+\sqrt q*\sqrt q)$
$(\sqrt {p^2}+\sqrt{pq}+\sqrt {pq}+\sqrt {q^2})+(\sqrt {p^2}-\sqrt{pq}-\sqrt {pq}+\sqrt {q^2})$
$(2\sqrt {p^2}+2\sqrt {q^2})$
$x^2=(2\sqrt {p^2}+2\sqrt {q^2})$
$\sqrt {x^2} = \sqrt {(2\sqrt {p^2}+2\sqrt {q^2})}$
