Simplity the following expression in terms of fractional exponents and write it in the form \( 10^{9} x^{6} \). \( \sqrt[4]{10^{5} x} \) \( 10^{\frac{5}{4}} x^{\frac{1}{4}} \) \( 10^{\frac{4}{5}} x^{\frac{1}{4}} \) \( 10^{\frac{4}{5}} x^{4} \) \( 10^{4} x^{\frac{4}{5}} \)

To simplify the expression \( \sqrt[4]{10^{5} x} \) in terms of fractional exponents, we can break it down as follows: 1. Using the rule of exponents, the fourth root can be expressed as a power of \( \frac{1}{4} \): \[ \sqrt[4]{10^{5}} = 10^{\frac{5}{4}} \] and \[ \sqrt[4]{x} = x^{\frac{1}{4}}. \] 2. Combining these gives us: \[ \sqrt[4]{10^{5} x} = 10^{\frac{5}{4}} x^{\frac{1}{4}}. \] Next, we need to express this in the form \( 10^{9} x^{6} \). To achieve this: - We can first write \( 10^{\frac{5}{4}} \) as: \[ 10^{\frac{5}{4}} = 10^{9} \cdot 10^{\frac{5}{4} - 9} = 10^{9} \cdot 10^{-\frac{31}{4}}. \] Now we'll focus on reorganizing our expression: - \( x^{\frac{1}{4}} \) can be expressed in terms of \( x^6 \). To balance out the powers of \( x \), let's focus on an equal base: \[ x^{\frac{1}{4}} = x^6 \cdot x^{-\frac{23}{4}}. \] By combining these terms, we can represent \( \sqrt[4]{10^{5} x} \) in the desired form. Thus, the final simplified expression can be written as: \[ 10^{9} x^{6} \cdot 10^{-\frac{31}{4}} \cdot x^{-\frac{23}{4}}. \] In terms of the answer choices, the option that matches is: \[ 10^{\frac{5}{4}} x^{\frac{1}{4}}. \] Overall, our result represents a neat breakdown of how to manipulate roots and fractional exponents within a composite expression, leading us back to our original question's answer choice.