Question

reduza a expressão a sua forma mais simples.

A
$\sqrt{50} - 3 \sqrt{2}$
B
$7 \sqrt{3} + \sqrt{12}$
C
$\sqrt{20} - \sqrt{45}$

​

Answer 1

Pra simplificar as expressões dadas, vamos reescrever os radicandos na sua forma fatorada e, posteriormente, aplicar as duas propriedades dos radicais listadas abaixo.

$(1)~~~\sqrt[b]{a\cdot c}~=~\sqrt[b]{a}~\cdot\sqrt[b]{c}\\\\(2)~~~\sqrt[b]{a^c}~=~a^{\frac{c}{b}}$

a)

$\sqrt{50}~-~3\sqrt{2}~=\\\\\\=~\sqrt{2\cdot5\cdot5}~-~3\sqrt{2}\\\\\\=~\sqrt{2\cdot5^2}~-~3\sqrt{2}\\\\\\=~\sqrt{2}\cdot\sqrt{5^2}~-~3\sqrt{2}\\\\\\=~\sqrt{2}\cdot5^{\frac{2}{2}}~-~3\sqrt{2}\\\\\\=~\sqrt{2}\cdot5^1~-~3\sqrt{2}\\\\\\=~5\sqrt{2}~-~3\sqrt{2}\\\\\\=~\boxed{2\sqrt{2}}$

b)

$7\sqrt{3}~+~\sqrt{12}~=\\\\\\=~7\sqrt{3}~+~\sqrt{2\cdot2\cdot3}\\\\\\=~7\sqrt{3}~+~\sqrt{2^2\cdot3}\\\\\\=~7\sqrt{3}~+~\sqrt{2^2}\cdot\sqrt{3}\\\\\\=~7\sqrt{3}~+~2^{\frac{2}{2}}\cdot\sqrt{3}\\\\\\=~7\sqrt{3}~+~2^1\cdot\sqrt{3}\\\\\\=~7\sqrt{3}~+~2\cdot\sqrt{3}\\\\\\=~\boxed{9\sqrt{3}}$

c)

$\sqrt{20}~-~\sqrt{45}~=\\\\\\=~\sqrt{2\cdot2\cdot5}~-~\sqrt{3\cdot3\cdot5}\\\\\\=~\sqrt{2^2\cdot5}~-~\sqrt{3^2\cdot5}\\\\\\=~\sqrt{2^2}\cdot\sqrt{5}~-~\sqrt{3^2}\cdot\sqrt{5}\\\\\\=~2^{\frac{2}{2}}\cdot\sqrt{5}~-~3^{\frac{2}{2}}\cdot\sqrt{5}\\\\\\=~2^1\cdot\sqrt{5}~-~3^1\cdot\sqrt{5}\\\\\\=~\boxed{-\sqrt{5}}$

Answer 2

Oie, Td Bom?!

A)

$\sqrt{50} - 3 \sqrt{2}$

$\sqrt{5 {}^{2} \times 2} - 3 \sqrt{2}$

$\sqrt{5 { }^{2} } \sqrt{2} - 3 \sqrt{2}$

$5 \sqrt{2} - 3 \sqrt{2}$

$(5 - 3) \sqrt{2}$

$2 \sqrt{2}$

B)

$7 \sqrt{3} + \sqrt{12}$

$7 \sqrt{3} + \sqrt{2 {}^{2} \times 3}$

$7 \sqrt{3} + \sqrt{2 {}^{2} } \sqrt{3}$

$7 \sqrt{3} + 2 \sqrt{3}$

$(7 + 2) \sqrt{3}$

$9 \sqrt{3}$

C)

$\sqrt{20} - \sqrt{45}$

$\sqrt{2 {}^{2} \times 5} - \sqrt{45}$

$\sqrt{2 {}^{2} } \sqrt{5} - \sqrt{45}$

$2 \sqrt{5} - \sqrt{45}$

$2 \sqrt{5} - \sqrt{3 {}^{2} \times 5}$

$2 \sqrt{5} - \sqrt{3 {}^{2} } \sqrt{5}$

$2 \sqrt{5} - 3 \sqrt{5}$

$(2 - 3) \sqrt{5}$

$- 1 \sqrt{5}$

$- \sqrt{5}$

Ajudei?! Dúvidas?! É Só Chamar!