Question

Simplifique as expressões a seguir :

Answer 1

Resposta:

a) $1$

b) $5^{9-\sqrt{6}}$

c) 16

d) 32768

Explicação passo-a-passo:

Letra a)

$\dfrac{9^\sqrt{2}}{3^\sqrt{8}} = \dfrac{9^\sqrt{2}}{3^{2\sqrt{2}}} = \dfrac{9^\sqrt{2}}{9^\sqrt{2}} = 1$

Letra b)

$\left(\dfrac{5^{\sqrt{2}+\sqrt{3}}}{25^{\sqrt{2}-\sqrt{3}}} \right)^{\sqrt{3}} = \left(\dfrac{5^{\sqrt{2}+\sqrt{3}}}{5^{2\sqrt{2}-2\sqrt{3}}} \right)^{\sqrt{3}} = (5^{-\sqrt{2} + 3\sqrt{3}})^{\sqrt{3}} = 5^{-\sqrt{6}+3\sqrt{9}} = 5^{9-\sqrt{6}}$

Letra c)

$\left(\dfrac{4^{\sqrt{5}}}{8^{\sqrt{20}}}\right)^{-\frac{1}{\sqrt{5}}} = \left(\dfrac{4^{\sqrt{5}}}{8^{2\sqrt{5}}}\right)^{-\frac{1}{\sqrt{5}}} = \left(\dfrac{2^{2\sqrt{5}}}{2^{6\sqrt{5}}}\right)^{-\frac{1}{\sqrt{5}}} = (2^{-4\sqrt{5}})^{-\frac{1}{\sqrt{5}}} = 2^{\frac{4\sqrt{5}}{\sqrt{5}}} = 2^4 = 16$

Letra d)

$\left(\dfrac{2^{\sqrt{27}}\cdot 8^{\sqrt{75}}}{4^{\sqrt{48}}}\right)^{\frac{\sqrt{3}}{2}} =\left(\dfrac{2^{3\sqrt{3}}\cdot 8^{5\sqrt{3}}}{4^{4\sqrt{3}}}\right)^{\frac{\sqrt{3}}{2}} = \left(\dfrac{2^{3\sqrt{3}}\cdot 2^{15\sqrt{3}}}{2^{8\sqrt{3}}}\right)^{\frac{\sqrt{3}}{2}} =$

$\left(\dfrac{2^{18\sqrt{3}}}{2^{8\sqrt{3}}}\right)^{\frac{\sqrt{3}}{2}} = \left(2^{10\sqrt{3}}\right)^{\frac{\sqrt{3}}{2}} = 2^{\frac{10\sqrt{9}}{2}} = 2^{15} = 32768$