QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
\sqrt[3]{70 n}(\sqrt[4]{70 n})^{2}
For what value of $x$ is the given expression equivalent to $(70 n)^{20 x}$, where $n>1$ ?
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Answer
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Step 1:: Recognize that the given expression is a cube root and a fourth power multiplied together.
\sqrt[3]{70 n} \times (\sqrt[4]{70 n})^2
Step 2:: To make the base of both radicals the same, we can rewrite the fourth root as a cube root and a remaining fourth power.
(\sqrt[3]{70 n}) \times (70 n)^{\frac{2}{4}}
Step 3:: Simplify the expression further.
(\sqrt[3]{70 n}) \times (70 n)^{\frac{1}{2}}
Step 4:: We want the exponent of $70 n$ to be $20x$.
(\sqrt[3]{70 n}) \times (70 n)^{20x - \frac{1}{3}}
Step 5:: Since we want the entire expression to be equal to $(70 n)^{20x}$, the exponents on the left side of the equation should be equal to the exponent on the right side of the equation.
\frac{1}{2} = 20x - \frac{1}{3}
Therefore, we have:
Step 6:: Solve for $x$.
x = \frac{5}{120}
Final Answer
The given expression is equivalent to $(70 n)^{20x}$ when $x = \boxed{\frac{1}{24}}$.
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