Question

07) Aplicando a regra dos produtos notáveis, desenvolva:
a)
$( \sqrt{5} + \sqrt[2]{3} ) ^{2}$
b)
$(2 - \sqrt{2} ) ^{2}$
c)
$( \sqrt{7} + \sqrt{5}).( \sqrt{7} - \sqrt{5} )$
d)
$(a + \sqrt{b} ) ^{2}$
e)
$(a + \sqrt{b} ) ^{2}$
f)
$x ^{2} - {y}^{2}$
g)
$( \sqrt{3} - 1) ^{2}$

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Answer 1

Explicação passo-a-passo:

a)

$\sf (\sqrt{5}+\sqrt{3})^2=(\sqrt{5})^2+2\cdot\sqrt{5}\cdot\sqrt{3}+(\sqrt{3})^2$

$\sf (\sqrt{5}+\sqrt{3})^2=5+2\cdot\sqrt{15}+3$

$\sf (\sqrt{5}+\sqrt{3})^2=5+3+2\sqrt{15}$

$\sf \red{(\sqrt{5}+\sqrt{3})^2=8+2\sqrt{15}}$

b)

$\sf (2-\sqrt{2})^2=2^2-2\cdot2\cdot\sqrt{2}+(\sqrt{2})^2$

$\sf (2-\sqrt{2})^2=4-4\sqrt{2}+2$

$\sf (2-\sqrt{2})^2=4+2-4\sqrt{2}$

$\sf \red{(2-\sqrt{2})^2=6-4\sqrt{2}}$

c)

$\sf (\sqrt{7}+\sqrt{5})\cdot(\sqrt{7}-\sqrt{5})$

$\sf =(\sqrt{7})^2-(\sqrt{5})^2$

$\sf =7-5$

$\sf =\red{2}$

d)

$\sf (a+\sqrt{b})^2=a^2+2\cdot a\cdot\sqrt{b}+(\sqrt{b})^2$

$\sf (a+\sqrt{b})^2=a^2+2a\sqrt{b}+b$

$\sf \red{(a+\sqrt{b})^2=a^2+b+2a\sqrt{b}}$

e)

$\sf (a+\sqrt{b})^2=a^2+2\cdot a\cdot\sqrt{b}+(\sqrt{b})^2$

$\sf (a+\sqrt{b})^2=a^2+2a\sqrt{b}+b$

$\sf \red{(a+\sqrt{b})^2=a^2+b+2a\sqrt{b}}$

f)

$\sf \red{x^2-y^2=(x+y)\cdot(x-y)}$

g)

$\sf (\sqrt{3}-1)^2=(\sqrt{3})^2-2\cdot\sqrt{3}\cdot1+1^2$

$\sf (\sqrt{3}-1)^2=3-2\sqrt{3}+1$

$\sf (\sqrt{3}-1)^2=3+1-2\sqrt{3}$

$\sf \red{(\sqrt{3}-1)^2=4-2\sqrt{3}}$