Given \( x>0 \), the expression \( \sqrt[4]{x^{29}} \) is equivalent to Answer \( x^{8} \) \( x^{7} \sqrt[4]{x} \) \( x^{7} \) \( x^{7} \sqrt[4]{x^{2}} \)

To simplify the expression \( \sqrt[4]{x^{29}} \), we can rewrite it using the property of exponents: \( \sqrt[4]{x^{29}} = x^{29/4} \). Dividing 29 by 4 gives us \( 7.25 \), which can be expressed as \( 7 + \frac{1}{4} \). Therefore, we have: \[ x^{29/4} = x^7 \cdot x^{1/4} = x^7 \sqrt[4]{x}. \] Thus, the expression simplifies to \( x^{7} \sqrt[4]{x} \). For the real-world application, this type of exponent manipulation is often used in fields such as physics and engineering where variables represent quantities. For instance, when dealing with formulas in mechanics or thermodynamics, simplifying expressions like these helps to solve equations more efficiently. Understanding these concepts allows engineers to model systems and predict behaviors accurately!