Examen online: Suma y resta de raíces

Falso, las partes de una raíz son índice, radical, radicando y raíz.

Falso, las raíces resultantes de una descomposición son equivalentes.

$$ 8\sqrt[7]{3} + 10\sqrt[7]{3} – 5\sqrt[7]{3} $$

$$ 18\sqrt[7]{3} – 5\sqrt[7]{3} $$

$$ \boxed{13\sqrt[7]{3}} $$

$$ \left( 9\sqrt{6} + 2\sqrt{6} \right) – \left( \sqrt{6} + 7\sqrt{6} \right) $$

$$ 11\sqrt{6} – 8\sqrt{6} $$

$$ \boxed{3\sqrt{6}} $$

Descomponiendo las raíces

• \( \sqrt[3]{875} = \sqrt[3]{125 \cdot 7} = \sqrt[3]{5^{3} \cdot 7} = \sqrt[3]{5^{3}} \cdot \sqrt[3]{7} = 5\sqrt[3]{7} \)
• \( \sqrt[3]{189} = \sqrt[3]{27 \cdot 7} = \sqrt[3]{3^{3} \cdot 7} = \sqrt[3]{3^{3}} \cdot \sqrt[3]{7} = 3\sqrt[3]{7} \)

Reemplazar y resolver

$$ \sqrt[3]{875} + 9\sqrt[3]{7} – \sqrt[3]{189} $$

$$ 5\sqrt[3]{7} + 9\sqrt[3]{7} – 3\sqrt[3]{7} $$

$$ 14\sqrt[3]{7} – 3\sqrt[3]{7} $$

$$ \boxed{11\sqrt[3]{7}} $$

Descomponiendo las raíces

• \( \sqrt{180} = \sqrt{36 \cdot 5} = \sqrt{6^{2} \cdot 5} = \sqrt{6^{2}} \cdot \sqrt{5} = 6\sqrt{5} \)
• \( \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{3^{2} \cdot 5} = \sqrt{3^{2}} \cdot \sqrt{5} = 3\sqrt{5} \)
• \( \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{2^{2} \cdot 5} = \sqrt{2^{2}} \cdot \sqrt{5} = 2\sqrt{5} \)

Reemplazar y resolver

$$ \sqrt{180} – \left( \sqrt{45} + \sqrt{20} \right) $$

$$ 6\sqrt{5} – \left( 3\sqrt{5} + 2\sqrt{5} \right) $$

$$ 6\sqrt{5} – 5\sqrt{5} $$

$$ \boxed{\sqrt{5}} $$

Descomponiendo las raíces

• \( \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{2^{2} \cdot 2} = \sqrt{2^{2}} \cdot \sqrt{2} = 2\sqrt{2} \)
• \( \sqrt{128} = \sqrt{64 \cdot 2} = \sqrt{8^{2} \cdot 2} = \sqrt{8^{2}} \cdot \sqrt{2} = 8\sqrt{2} \)
• \( \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{4^{2} \cdot 2} = \sqrt{4^{2}} \cdot \sqrt{2} = 4\sqrt{2} \)

Reemplazar y resolver

$$ \left[ \sqrt{8} – \left( -\sqrt{2} \right) \right] + \left[ \left( -\sqrt{128} \right) – \sqrt{32} \right] $$

$$ \left[ 2\sqrt{2} – \left( -\sqrt{2} \right) \right] + \left[ \left( -8\sqrt{2} \right) – 4\sqrt{2} \right] $$

$$ \left[ 2\sqrt{2} + \sqrt{2} \right] + \left[ \left( -8\sqrt{2} \right) – 4\sqrt{2} \right] $$

$$ 3\sqrt{2} + \left( -12\sqrt{2} \right) $$

$$ \boxed{-9\sqrt{2}} $$

Descomponiendo las raíces

• \( \sqrt[4]{1875} = \sqrt[4]{625 \cdot 3} = \sqrt[4]{5^{4} \cdot 3} = \sqrt[4]{5^{4}} \cdot \sqrt[4]{3} = 5\sqrt[4]{3} \)
• \( \sqrt[4]{48} = \sqrt[4]{16 \cdot 3} = \sqrt[4]{2^{4} \cdot 3} = \sqrt[4]{2^{4}} \cdot \sqrt[4]{3} = 2\sqrt[4]{3} \)
• \( \sqrt[4]{768} = \sqrt[4]{256 \cdot 3} = \sqrt[4]{4^{4} \cdot 3} = \sqrt[4]{4^{4}} \cdot \sqrt[4]{3} = 4\sqrt[4]{3} \)

Reemplazar y resolver

$$ \left( \sqrt[4]{1875} – 2\sqrt[4]{48} \right) + 5\sqrt[4]{768} $$

$$ \left[ 5\sqrt[4]{3} – 2\left( 2\sqrt[4]{3} \right) \right] + 5\left( 4\sqrt[4]{3} \right) $$

$$ \left[ 5\sqrt[4]{3} – 4\sqrt[4]{3} \right] + 20\sqrt[4]{3} $$

$$ \sqrt[4]{3} + 20\sqrt[4]{3} $$

$$ \boxed{21\sqrt[4]{3}} $$