## Algebra 1

a) $3\sqrt2$ b) $2\sqrt7$ c) $\sqrt {(2\sqrt {p^2}+2\sqrt {q^2})}$
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) $(\sqrt {p}+\sqrt q)^2+ (\sqrt {p}-\sqrt q)^2=x^2$ $(\sqrt {p}*\sqrt{p}+\sqrt{p}*\sqrt q+\sqrt q*\sqrt{p}+\sqrt q*\sqrt q)+(\sqrt {p}*\sqrt{p}-\sqrt{p}*\sqrt q-\sqrt q*\sqrt{p}+\sqrt q*\sqrt q)$ $(\sqrt {p^2}+\sqrt{pq}+\sqrt {pq}+\sqrt {q^2})+(\sqrt {p^2}-\sqrt{pq}-\sqrt {pq}+\sqrt {q^2})$ $(2\sqrt {p^2}+2\sqrt {q^2})$ $x^2=(2\sqrt {p^2}+2\sqrt {q^2})$ $\sqrt {x^2} = \sqrt {(2\sqrt {p^2}+2\sqrt {q^2})}$