Question

1)fatore o número que aparece no radicando,e a seguir,simplifique cada um dos radicais retirando fatores do radicando.

2)Sabendo que
$x = \sqrt{2304} \: e \: y = \sqrt {}^{6} {64}$
Qual é o valor da razão?
$\frac{x}{y}$

3)Transforme as expressões em um só radical e depois,calcule o valor de cada uma.

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Answer 1

2)Sabendo que

$\sqrt{2304} ~~e~~y=\sqrt[6]{64} \\\\\\Fatorando\\\\2304/2\\\\1152/2\\\\~~576/2\\\\288/2\\\\144/2\\\\72/2\\\\36/2\\\\18/2\\\\9/3\\\\3/3\\\\1\\\\Ficando:\\\\\sqrt{2^8.3^2} (Numerador|)\\\\\\Denominador\\\\\sqrt[6]{64} \\\\Fatorando:\\\\64/2\\\\32/2$

$16/2\\\\8/2\\\\4/2\\\\2/2\\\\1\\Ficando:\\\\\sqrt[6]{2^6} \\\\\\X=\sqrt{2^8.3^2} \\\\y=\sqrt[6]{2^6} \\\\Pergunta:\dfrac{x}{y} \\\\\\Logo, ~~temos:\\\\\\=\dfrac{\sqrt{2^8.3^2} }{\sqrt[6]{2^6} } \\\\\\=\dfrac{\sqrt{2^2.2^2.2^2.2^2.3^2} }{\sqrt[6]{2^6} } \\\\\\=\dfrac{2.2.2.2.3}{6} \\\\\\=\dfrac{48}{6} \\\\\\Resposta=8$

3)Transforme as expressões em um só radical e depois,calcule o valor de cada uma.

$a)\sqrt{75} \\\\Fatorando,~ temos:\\75/5\\15/5\\3/3\\1\\\\\sqrt{5^2.3 } \\\\Resposta=5\sqrt{3} \\\\\\b)\sqrt{700}\\\\Fatorando$

$700/2\\350/2\\175/5\\35/5\\7/7\\1\\\\=\sqrt{2^2.7.5^2} \\\\\\Resposta=10\sqrt{7}$

$c)\sqrt[3]{250} \\\\Fatorando:\\\\250/2\\125/5\\25/5\\5/5\\1\\\\Resposta=\sqrt[3]{2.5^3} } \\\\\\Resposta=5\sqrt[3]{2} \\\\\\d)\sqrt[5]{192} \\\\Fatorando\\192/2\\96/2\\48/2\\24/2\\12/2\\6/2\\3/3\\1\\=\sqrt[5]{2^5.3} \\\\Resposta=2\sqrt[5]{3} \\\\\\e)\sqrt[4]{176} \\\\Fatorando\\\\176/2\\88/2\\44/2\\22/2\\11/11\\1\\\\=\sqrt[4]{2^4.11}$

$Resposta=4\sqrt[4]{11} \\\\\\f)\sqrt{800} \\\\Fatorando:\\\\800/2\\400/2\\200/2\\100/2\\50/2\\25/5\\5/5\\1\\\\\\=\sqrt{2^2.2^2.2^1.5^2} \\\\Resposta=20\sqrt{2} \\\\\\g)\sqrt{1800} \\\\Fatorando:\\\\1800/2\\900/2\\450/2\\225/5\\45/5\\9/3\\3/3\\1\\\\=\sqrt{2^2.2.3^2.5^2} \\\\Resposta=30\sqrt{2} \\\\\\h)\sqrt[3]{375} \\\\Fatorando:$

$375/3\\125/5\\25/5\\5/5\\1\\\\=\sqrt[3]{3.5^3} \\\\Resposta=5\sqrt[3]{3} \\\\\\i)\sqrt{2700} \\\\Fatorando:\\\\2700/2\\1350/2\\675/5\\135/5\\27/3\\9/3\\3/3\\1\\\\=\sqrt{2^2.5^2.3^2.3^1} \\\\\\Resposta=30\sqrt{3} \\\\\\j)\sqrt[6]{640} \\\\Fatorando:\\640/2\\320/2\\160/2\\80/2\\40/2\\20/2\\10/2\\5/5\\1\\\\=\sqrt[6]{2^6.2^1.5^1} \\\\Resposta=2\sqrt[6]{10}$

5)

$a)\sqrt{\sqrt[3]{\sqrt{4096} } } \\\\Fatorando\\\\4096/2\\2048/2\\1024/2\\512/2\\256/2\\128/2\\64/2\\32/2\\16/2\\8/2\\4/2\\2/2\\1\\\\=\sqrt{\sqrt[3]{\sqrt{2^1^2} } } \\\\\sqrt[2.3.2]{2^1^2} \\\\=\sqrt[12]{2^1^2} \\\\\\Resposta=2$

$b)\sqrt{\sqrt{10000} } \\\\\\\sqrt[2.2]{10000} \\\\\\Fatorando:\\\\10000/2\\5000/2\\2500/2\\1250/2\\625/5\\125/5\\25/5\\5/5\\1\\\\=\sqrt[4]{2^4.5^4} \\\\\\Resposta=10$