Question

CALCULE AS POTENCIAS;a)(√15)elevado a 2;b)(√7+√3)elevado a 2;c)(3√7)elevado a 2;d)(3-7)elevado a 2;e)(√2+√3)elevado a 2

Answer 1

vou representar os expoentes com '^' antes. Ex: 2 elevado a 3 = 2^3
vou usar 'x' como sinal de multiplicação

a) (√15)^2 = (15^1/2)^2 = 15
b)(√7 + √3)^2 = [(√7)^2 + 2 x √7 x √3 + (√3)^2] = 7 + 2√21 + 3 = 10 + 2√21 = 2(5 + √21)
c) (3√7)^2 = 9 x 7 = 63
d) (3-7)^2 = (-4)^2 = 16
e) (√2 + √3)^2 = [(√2)^2 + 2 x √2 x √3 + (√3)^2] = 2 + 2√6 + 3 = 5 + 2√6

Answer 2

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$\boxed{\begin{array}{l}\tt a)~\sf(\sqrt{15})^2=\sqrt{15^{\backslash\!\!\!2}}=15\\\tt b)~\sf(\sqrt{7}+\sqrt{3})^2=(\sqrt{7})^2+2\sqrt{7}\cdot\sqrt{3}+(\sqrt{3})^2\\\sf(\sqrt{7}+\sqrt{3})^2=7+2\sqrt{21}+3=10+2\sqrt{21}\\\tt c )~\sf(3\sqrt{7})^2=3^2\cdot(\sqrt{7})^2=9\cdot7=63\\\tt d)~\sf(3-7)^2=(-4)^2=16\\\tt e)~\sf(\sqrt{2}+\sqrt{3})^2=(\sqrt{2})^2+2\cdot\sqrt{2}\cdot\sqrt{3}+(\sqrt{3})^2\\\sf (\sqrt{2}+\sqrt{3})^2=2+2\sqrt{6}+3\\\sf (\sqrt{2}+\sqrt{3})^2=5+2\sqrt{6}\end{array}}$

$\Large\boxed{\begin{array}{l}\underline{\rm de~modo~geral\!:}\\\sf(\sqrt{a}+\sqrt{b})^2=a+b+2\sqrt{ab}\,\forall a,b\in\mathbb{R}\end{array}}$