We appreciate your visit to Simplify tex sqrt 3 5x cdot sqrt 3 25x 2 tex completely A tex 25x 3 tex B tex 25x tex C tex 5x 3. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
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.
Thanks for taking the time to read Simplify tex sqrt 3 5x cdot sqrt 3 25x 2 tex completely A tex 25x 3 tex B tex 25x tex C tex 5x 3. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
