Which is equivalent toStartRoot 10 EndRoot Superscript three-fourths x ?
RootIndex 3 StartRoot 10 EndRoot Superscript 4 x
RootIndex 4 StartRoot 10 EndRoot Superscript 3 x
RootIndex 5 StartRoot 10 EndRoot Superscript 4 x
RootIndex 8 StartRoot 10 EndRoot Superscript 3 x
The Correct Answer and Explanation is :
To determine which expression is equivalent to ( \sqrt[4]{10^{\frac{3}{4}} x} ), we can use properties of exponents and roots.
Step-by-Step Breakdown
- Understanding the Expression:
The expression ( \sqrt[4]{10^{\frac{3}{4}} x} ) can be rewritten using exponent notation:
[
\sqrt[4]{10^{\frac{3}{4}} x} = (10^{\frac{3}{4}} x)^{\frac{1}{4}} = 10^{\frac{3}{4} \cdot \frac{1}{4}} x^{\frac{1}{4}} = 10^{\frac{3}{16}} x^{\frac{1}{4}}.
] - Simplifying the Terms:
From the simplification above, we see that:
[
\sqrt[4]{10^{\frac{3}{4}} x} = 10^{\frac{3}{16}} \cdot x^{\frac{1}{4}}.
] - Comparing with Given Options:
Now, let’s analyze the provided options:
- Option 1: ( \sqrt[3]{10^{4} x} )
[
= (10^4 x)^{\frac{1}{3}} = 10^{\frac{4}{3}} x^{\frac{1}{3}}.
] - Option 2: ( \sqrt[4]{10^{3} x} )
[
= (10^3 x)^{\frac{1}{4}} = 10^{\frac{3}{4}} x^{\frac{1}{4}}.
] - Option 3: ( \sqrt[5]{10^{4} x} )
[
= (10^4 x)^{\frac{1}{5}} = 10^{\frac{4}{5}} x^{\frac{1}{5}}.
] - Option 4: ( \sqrt[8]{10^{3} x} )
[
= (10^3 x)^{\frac{1}{8}} = 10^{\frac{3}{8}} x^{\frac{1}{8}}.
]
- Finding the Equivalent Expression:
To find an equivalent expression to ( 10^{\frac{3}{16}} x^{\frac{1}{4}} ), we observe the exponents of 10 and ( x ) in the simplified expressions:
- None of the options have the exponent ( \frac{3}{16} ) for 10 or ( \frac{1}{4} ) for ( x ).
Conclusion
After evaluating all options, we conclude that none of the provided options are equivalent to ( \sqrt[4]{10^{\frac{3}{4}} x} ).
Correct Expression
The equivalent expression in this context can be represented as ( 10^{\frac{3}{16}} x^{\frac{1}{4}} ), which is not listed among the options. Therefore, it is crucial to ensure the context or values are correctly interpreted or restated to find an appropriate match.