Answer :
Certainly! Let's simplify the expression [tex]\(\frac{\sqrt[3]{5x}}{\sqrt[3]{25x^2}}\)[/tex].
### Step-by-step Solution
1. Write the Expression in Terms of Fractional Exponents:
- [tex]\(\sqrt[3]{5x}\)[/tex] can be written as [tex]\((5x)^{\frac{1}{3}}\)[/tex].
- [tex]\(\sqrt[3]{25x^2}\)[/tex] can be written as [tex]\((25x^2)^{\frac{1}{3}}\)[/tex].
So, the expression becomes:
[tex]\[
\frac{(5x)^{\frac{1}{3}}}{(25x^2)^{\frac{1}{3}}}
\][/tex]
2. Combine the Exponents Using Properties of Exponents:
Using the rule [tex]\(\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m\)[/tex], we can combine the expression inside the cube roots:
[tex]\[
\left(\frac{5x}{25x^2}\right)^{\frac{1}{3}}
\][/tex]
3. Simplify the Fraction Inside the Cube Root:
[tex]\[
\frac{5x}{25x^2} = \frac{5}{25} \cdot \frac{x}{x^2}
\][/tex]
- [tex]\(\frac{5}{25} = \frac{1}{5}\)[/tex]
- [tex]\(\frac{x}{x^2} = \frac{1}{x}\)[/tex]
So the fraction simplifies to:
[tex]\[
\frac{1}{5x}
\][/tex]
4. Apply the Cube Root:
Now, substitute back into the root expression:
[tex]\[
\left(\frac{1}{5x}\right)^{\frac{1}{3}} = \frac{1}{(5x)^{\frac{1}{3}}}
\][/tex]
Simplifying the cube root: [tex]\( (5x)^{\frac{1}{3}}\)[/tex] is the same as [tex]\(\sqrt[3]{5x}\)[/tex].
The final simplified expression is:
[tex]\[
\frac{1}{\sqrt[3]{5x}}
\][/tex]
The correct choice is not a straightforward match to the options provided, suggesting there might be an oversight either in setting the options or in having an expectation of further manipulation. However, in terms of pure simplification, we reached [tex]\( \frac{1}{\sqrt[3]{5x}} \)[/tex]. If you need to express under a found option and a typical match through comprehension, sometimes equations use equivalency in evaluation at more simplistic qualitative intervals or specific known cases like [tex]\( f(x) = 1/x \)[/tex] satisfying choice conventions trapped at the next analysis point.
If the options provided were alongside expected formatted workings compatible to practical school tasking statuses, more information may have indicated extra constraints or visible expectations - none achieved thus purely from given choices until we've re-aligned or clarified task expectations known potentially academically non-naturally predictively fitting set outputs.
### Step-by-step Solution
1. Write the Expression in Terms of Fractional Exponents:
- [tex]\(\sqrt[3]{5x}\)[/tex] can be written as [tex]\((5x)^{\frac{1}{3}}\)[/tex].
- [tex]\(\sqrt[3]{25x^2}\)[/tex] can be written as [tex]\((25x^2)^{\frac{1}{3}}\)[/tex].
So, the expression becomes:
[tex]\[
\frac{(5x)^{\frac{1}{3}}}{(25x^2)^{\frac{1}{3}}}
\][/tex]
2. Combine the Exponents Using Properties of Exponents:
Using the rule [tex]\(\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m\)[/tex], we can combine the expression inside the cube roots:
[tex]\[
\left(\frac{5x}{25x^2}\right)^{\frac{1}{3}}
\][/tex]
3. Simplify the Fraction Inside the Cube Root:
[tex]\[
\frac{5x}{25x^2} = \frac{5}{25} \cdot \frac{x}{x^2}
\][/tex]
- [tex]\(\frac{5}{25} = \frac{1}{5}\)[/tex]
- [tex]\(\frac{x}{x^2} = \frac{1}{x}\)[/tex]
So the fraction simplifies to:
[tex]\[
\frac{1}{5x}
\][/tex]
4. Apply the Cube Root:
Now, substitute back into the root expression:
[tex]\[
\left(\frac{1}{5x}\right)^{\frac{1}{3}} = \frac{1}{(5x)^{\frac{1}{3}}}
\][/tex]
Simplifying the cube root: [tex]\( (5x)^{\frac{1}{3}}\)[/tex] is the same as [tex]\(\sqrt[3]{5x}\)[/tex].
The final simplified expression is:
[tex]\[
\frac{1}{\sqrt[3]{5x}}
\][/tex]
The correct choice is not a straightforward match to the options provided, suggesting there might be an oversight either in setting the options or in having an expectation of further manipulation. However, in terms of pure simplification, we reached [tex]\( \frac{1}{\sqrt[3]{5x}} \)[/tex]. If you need to express under a found option and a typical match through comprehension, sometimes equations use equivalency in evaluation at more simplistic qualitative intervals or specific known cases like [tex]\( f(x) = 1/x \)[/tex] satisfying choice conventions trapped at the next analysis point.
If the options provided were alongside expected formatted workings compatible to practical school tasking statuses, more information may have indicated extra constraints or visible expectations - none achieved thus purely from given choices until we've re-aligned or clarified task expectations known potentially academically non-naturally predictively fitting set outputs.
