Question

Determine o valor do número x em cada uma das igualdades. a)¹⁴√2⁸=x√2⁴ b) ¹⁵√10=³√10x c) ⁸√5⁴=√5x d) ¹⁰√6x=⁵√6x ​


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Answer 1

Resposta:

$a:\ x=0,37149\\\\b:\ x=0,158489\\\\c:\ x=125\\\\d:\ x=0\ e\ x=\frac{1}{6}$

Explicação passo a passo:

$a)\ \sqrt[14]{2^8}=x\sqrt{2^4}\\\\\sqrt[14]{256}=x\sqrt{2^2\bullet2^2}\\\\\sqrt[14]{256}=x\bullet2\bullet2\\\\\sqrt[14]{256}=4x\\\\x=\frac{\sqrt[14]{256}}{4}\\\\x=\frac{1,48599}{4}\\\\x=0,37149 \\\\\\b)\ \sqrt[15]{10}=\sqrt[3]{10x}\\\\(\sqrt[15]{10})^3=(\sqrt[3]{10x})^3\\\\(10^{\frac{1}{15} })^3=((10x)^{\frac{1}{3}})^3\\\\10^{\frac{3}{15}}=(10x)^{\frac{3}{3}}\\\\10^{\frac{1}{5}}=10x\\\\\sqrt[5]{10}=10x\\\\x=\frac{\sqrt[5]{10}}{10}\\\\x=\frac{1,58489}{10}\\\\x=0,158489$

$c)\ \sqrt[2]{5^4}=\sqrt{5x}\\\\(\sqrt[2]{5^4})^2=(\sqrt{5x})^2\\\\(5^{\frac{4}{2}})^2=[(5x)^{\frac{1}{2}}]^2\\\\5^{\frac{8}{2}}=(5x)^{\frac{2}{2}}\\\\5^4=5x\\\\5\bullet5\bullet5\bullet5=5x\\\\625=5x\\\\x=\frac{625}{5}\\\\x=125\\\\ou\\\\5^4=5x\\\\x=\frac{5^4}{5}\\\\x=5^{4-1}\\\\x=5^3\\\\x=125$

$d)\ \sqrt[10]{6x}=\sqrt[5]{6x}\\\\(\sqrt[10]{6x})^{10}=(\sqrt[5]{6x})^{10}\\\\((6x)^{\frac{1}{10}})^{10}=((6x)^{\frac{1}{5}})^{10}\\\\(6x)^{\frac{10}{10}}=(6x)^{\frac{10}{5}}\\\\6x=(6x)^2\\\\6x=36x^2\\\\36x^2-6x=0\\\\6x(x-6)=0\\\\x=0\ e\ x=\frac{1}{6}$