Question
2. Write in radical form: \( 2.1 x^{\frac{3}{5}} \) \( 226 x^{\frac{1}{4}} \) 2.3. \( \left(3 x^{2}\right)^{\frac{2}{3}} \)
Ask by Wang Murphy. in South Africa
Jan 22,2025
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Tutor-Verified Answer
Answer
Here are the expressions in radical form:
1. \( x^{\frac{3}{5}} = \sqrt[5]{x^{3}} \)
2. \( 226 x^{\frac{1}{4}} = 226 \sqrt[4]{x} \)
3. \( \left(3 x^{2}\right)^{\frac{2}{3}} = \sqrt[3]{9 x^{4}} \)
Solution
Certainly! Here are the expressions written in radical form:
### Problem 2
#### 2.1. \( x^{\frac{3}{5}} \)
\[
x^{\frac{3}{5}} = \sqrt[5]{x^{3}} = \left(\sqrt[5]{x}\right)^{3}
\]
#### 2.2. \( 226 x^{\frac{1}{4}} \)
\[
226 x^{\frac{1}{4}} = 226 \sqrt[4]{x}
\]
#### 2.3. \( \left(3 x^{2}\right)^{\frac{2}{3}} \)
\[
\left(3 x^{2}\right)^{\frac{2}{3}} = \sqrt[3]{\left(3 x^{2}\right)^{2}} = \sqrt[3]{9 x^{4}} = \left(\sqrt[3]{3 x^{2}}\right)^{2}
\]
---
**Summary of Radical Forms:**
1. \( x^{\frac{3}{5}} = \sqrt[5]{x^{3}} \)
2. \( 226 x^{\frac{1}{4}} = 226 \sqrt[4]{x} \)
3. \( \left(3 x^{2}\right)^{\frac{2}{3}} = \sqrt[3]{9 x^{4}} \)
Feel free to choose the form that best fits your needs!
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Simplify this solution Extra Insights
To express \( 2.1 x^{\frac{3}{5}} \) in radical form, you can write it as \( 2.1 \sqrt[5]{x^3} \). This means you take the fifth root of \( x^3 \) and multiply it by 2.1 for complete clarity. For \( 226 x^{\frac{1}{4}} \), you can rewrite it as \( 226 \sqrt[4]{x} \). Here, you're taking the fourth root of \( x \) and multiplying it by 226, keeping it neat and straightforward! Lastly, for \( (3 x^{2})^{\frac{2}{3}} \), this can be written as \( 3^{\frac{2}{3}} (x^2)^{\frac{2}{3}} \), which simplifies to \( 3^{\frac{2}{3}} x^{\frac{4}{3}} \). You can further express \( x^{\frac{4}{3}} \) as \( x^{4/3} = x^{4} \sqrt[3]{x} \), combining the powers for a tidy solution!
