Question

calcule o valor das expressões numéricas

Answer 1

a) 

$\sqrt[3]{5}\cdot\sqrt[3]{2}+\sqrt[3]{80}\\\\ \sqrt[3]{5\cdot \:2}+\sqrt[3]{80}\\\\ \sqrt[3]{10}+\sqrt[3]{80}$

Fatorando o número 80, temos que 80 = 2⁴ · 5

$\sqrt[3]{10}+\sqrt[3]{2^{4} \cdot 5}\\\\ \sqrt[3]{10}+\sqrt[3]{2^{4} \cdot 5}\\\\ usando\;\; a\;\; propriedade\;\;\:a^{n+m}=a^{n}\cdot a^{m}\\\\ \sqrt[3]{10}+\sqrt[3]{2^3\cdot \:2\cdot \:5}\\\\ \sqrt[3]{10}+\sqrt[3]{2^3}\sqrt[3]{2\cdot \:5}\\\\ \sqrt[3]{10}+\sqrt[\not{3}]{2^{\not{3}}}\sqrt[3]{2\cdot \:5}\\\\ \sqrt[3]{10}+ 2\cdot \sqrt[3]{2\cdot \:5}\\\\ \sqrt[3]{10}+ 2\cdot \sqrt[3]{10}\\\\ (1+2) \cdot \sqrt[3]{10}\\\\ 3\sqrt[3]{10}$


b)

$\left(\sqrt{98}-\sqrt{50}\right)^3$

Fatorando 98 e 50, obtemos:

98 = 2 · 7 ²
50 = 2 · 5²

$\left(\sqrt{2\cdot \:7^2}-\sqrt{2\cdot \:5^2}\right)^3\\\\ \left(\sqrt{2}\sqrt{7^2}}-\sqrt{2}\sqrt{5^2}\right)^3$

$\left(\sqrt{2}\cdot \:7}-\sqrt{2}\cdot \:5}\right)^3\\\\ \left((7-5) \cdot \sqrt{2}\right)^3\\\\ \left(2\sqrt{2}\right)^3\\\\ como\;\;\left(a\cdot \:b\right)^n=a^nb^n\\\\ 2^3\left(\sqrt{2}\right)^3 = 2^3\cdot \:2^{\frac{3}{2}} = 8 \cdot 2^{1+\frac{1}{2}} =\\\\ = 8\cdot 2^1\cdot \:2^{\frac{1}{2}} = 8 \cdot 2\sqrt{2} = \\\\ = 16 \sqrt{2}$