Question
Express \( x^{\frac{3}{4}} \) in terms of a radical expression.
Ask by Hodges Gough. in South Africa
Jan 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x^{\frac{3}{4}} \) can be written as the fourth root of \( x^3 \), which is \( \sqrt[4]{x^3} \).
Solution
To express \( x^{\frac{3}{4}} \) as a radical, you can use the relationship between exponents and roots. The general form is:
\[
x^{\frac{a}{b}} = \sqrt[b]{x^a}
\]
Applying this to \( x^{\frac{3}{4}} \):
\[
x^{\frac{3}{4}} = \sqrt[4]{x^3}
\]
**Alternative Forms:**
1. **Inside the Radical:**
\[
\sqrt[4]{x^3}
\]
This represents the fourth root of \( x^3 \).
2. **Outside the Radical:**
\[
\left( \sqrt[4]{x} \right)^3
\]
This represents the cube of the fourth root of \( x \).
Both forms are mathematically equivalent and correct. You can choose either based on the context or preference.
**Final Answer:**
\[
x^{\frac{3}{4}} = \sqrt[4]{\,x^{\,3\,}}
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
To express \( x^{\frac{3}{4}} \) in terms of a radical expression, we can rewrite the exponent as a square root. The fraction \(\frac{3}{4}\) indicates that we will take the fourth root and then cube the result. This can be written as: \[ x^{\frac{3}{4}} = \sqrt[4]{x^3} \] Thus, \( x^{\frac{3}{4}} \) in radical form is \( \sqrt[4]{x^3} \).
