Respuesta :
Si ∛(5^x )=1/∛25 . Calcula el valor de x es
[tex] \sqrt[3]{5^x}= \dfrac{1}{ \sqrt[3]{25} } \\ \\ \\ \sqrt[3]{5^x}= ({ \sqrt[3]{25} }) ^{-1} \\ \\ \\ \sqrt[3]{5^x}= ({ \sqrt[3]{5^2} }) ^{-1} \\ \\ \\ \sqrt[3]{5^x}= { \sqrt[3]{5} ^ {-2}} \\ \\ \\ (\sqrt[3]{5})^x= ({ \sqrt[3]{5}) ^ {-2}} \quad\to\quad \boxed{ x= -2} \\ \\ \\ [/tex]
[tex] \sqrt[3]{5^x}= \dfrac{1}{ \sqrt[3]{25} } \\ \\ \\ \sqrt[3]{5^x}= ({ \sqrt[3]{25} }) ^{-1} \\ \\ \\ \sqrt[3]{5^x}= ({ \sqrt[3]{5^2} }) ^{-1} \\ \\ \\ \sqrt[3]{5^x}= { \sqrt[3]{5} ^ {-2}} \\ \\ \\ (\sqrt[3]{5})^x= ({ \sqrt[3]{5}) ^ {-2}} \quad\to\quad \boxed{ x= -2} \\ \\ \\ [/tex]
The value of x in equation [tex]\sqrt[3]{{{5^x}}} = \dfrac{1}{{\sqrt[3]{{25}}}}[/tex] is [tex]\boxed{x = - 2}.[/tex]
Further Explanation:
The rules of exponents are as follows,
1. [tex]\boxed{\left( {{x^m}} \right) \times \left( {{x^n}} \right) = {x^{m + n}}}[/tex]
2. [tex]\boxed{\frac{{{x^m}}}{{{x^n}}} = {x^{m - n}}}[/tex]
3. [tex]\boxed{{{\left( {{x^a}} \right)}^b} = {x^{a \times b}}}[/tex]
4. [tex]\boxed{{x^{\frac{m}{n}}} = \sqrt[n]{{{x^m}}}}[/tex]
Given:
The equation is [tex]\sqrt[3]{{{5^x}}} = \dfrac{1}{{\sqrt[3]{{25}}}}.[/tex]
Calculation:
Solve the given equation [tex]\sqrt[3]{{{5^x}}} = \dfrac{1}{{\sqrt[3]{{25}}}}[/tex] to obtain the value of [tex]x[/tex].
[tex]\begin{aligned}\left({\sqrt[3]{{{5^x}}}} \right)&= \frac{1}{{\sqrt[3]{{25}}}}\\\sqrt[3]{{{5^x}}}&= \frac{1}{{\sqrt[3]{{{5^2}}}}}\\\sqrt[3]{{{5^x}}} &= {\left( {\sqrt[3]{{{5^2}}}} \right)^{ - 1}}\\\sqrt[3]{{{5^x}}}&= {\left({\sqrt[3]{5}} \right)^{ - 2}}\\\end{aligned}[/tex]
Further solve the above equation to obtain the value of [tex]x[/tex].
[tex]{\left( {\sqrt[3]{5}} \right)^x} = {\left( {\sqrt[3]{5}} \right)^{ - 2}}[/tex]
By comparing the two sides the value of [tex]x[/tex] is [tex]x = - 2.[/tex]
The value of x in equation [tex]\sqrt[3]{{{5^x}}} = \dfrac{1}{{\sqrt[3]{{25}}}}[/tex] is [tex]\boxed{x = - 2}.[/tex]
Learn more:
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Exponents and Powers
Keywords: value of [tex]x[/tex], cube root, solution, factorized form, exponents, power, equation, power rule, exponent rule, powers,
