Question

efectua las siguientes radicales​

Answer 1

Al resolver los radicales se obtiene aplicando:

Propiedad de los exponentes:

$(a^{b})^{c} = a^{b.c}$

Propiedades de radicales:

$\sqrt{x} = x^{\frac{1}{2}} \\\sqrt{x.y} = \sqrt{x} . \sqrt{y}$

$a)\sqrt{\sqrt[4]{256x^{8} } } \\=\sqrt{\sqrt[4]{256}\sqrt[4]{x^{8} } }\\=\sqrt{4 x^{\frac{8}{4} } }\\=\sqrt{4 x^{2} }\\=\sqrt{4} \sqrt{ x^{2}} \\= 2x$

$b) \sqrt{\sqrt[5]{225x^{6}y^{4}} } \\= [(225x^{6}y^{4})^{\frac{1}{5}}]^{\frac{1}{2}}\\= (225x^{6}y^{4})^{\frac{1}{10}}\\= \sqrt[10]{225} \sqrt[10]{x^{6}y^{4}}$

$c)\sqrt{3x\sqrt[3]{3\sqrt{3} } } \\=[3x(3(3^{\frac{1}{2}}))^{\frac{1}{3}} ]^{\frac{1}{2}} \\=[3x(3^{\frac{1}{3}})(3^{\frac{1}{6}})]^{\frac{1}{2}} \\= 3^{\frac{1}{2}}. x^{\frac{1}{2}}.3^{(\frac{1}{3}+\frac{1}{6})}\\=\sqrt{3}\sqrt{x\sqrt{3} }$

$d)\sqrt{32a+\sqrt{8a^{2}+\sqrt{64a^{4} } } } \\=\sqrt{32a+\sqrt{8a^{2}+\sqrt{64} a^{\frac{4}{2} } } } } \\=\sqrt{32a+\sqrt{8a^{2}+8 a^{2} } } }\\=\sqrt{32a+\sqrt{16a^{2}} } \\=\sqrt{32a+\sqrt{16}a} \\=\sqrt{32a+4a} \\=\sqrt{36a}\\=6\sqrt{a}$

$e)\sqrt{44+\sqrt{22+\sqrt{9}}}\\= \sqrt{44+\sqrt{22+3}}\\= \sqrt{44+\sqrt{25}} \\ =\sqrt{44+5} \\=\sqrt{49}\\= 7$

$f)\sqrt{40xy+\sqrt{80x^{2}y^{2}+\sqrt{x^{4}y^{4}}}} \\=\sqrt{40xy+\sqrt{80x^{2}y^{2}+x^{\frac{4}{2} }y^{\frac{4}{2}}}}} \\=\sqrt{40xy+\sqrt{80x^{2}y^{2}+x^{2}y^{2}}}}} \\=\sqrt{40xy+\sqrt{81x^{2}y^{2}}}}} \\=\sqrt{40xy+\sqrt{81} xy}}\\=\sqrt{40xy + 9xy}\\=\sqrt{49xy}}\\=7\sqrt{xy}$