Write the expression in radical form.
$x^{\frac{5}{4}}$
To convert the expression \(x^{\frac{5}{4}}\) into radical form, we rely on the fundamental definition of rational exponents. For a positive real number \(x\) and positive integers \(m\) and \(n\), an expression of the form \(x^{\frac{m}{n}}\) can be expressed as a radical using the property:
\[x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\]
In our case, the base is \(x\), the numerator \(m\) is 5, and the denominator \(n\) is 4. Substituting these values into the formula gives us the radical form.
Applying the identity \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\) with \(m=5\) and \(n=4\), we obtain:
\[x^{\frac{5}{4}} = \sqrt[4]{x^5}\]
This expression signifies the fourth root of \(x\) raised to the fifth power. We can also write this as the fourth root of \(x^4 \times x\). Since \(x^4\) is a perfect fourth power, we can simplify the expression further by pulling it out of the radical:
\[\sqrt[4]{x^4 \cdot x} = \sqrt[4]{x^4} \cdot \sqrt[4]{x} = x\sqrt[4]{x}\]
Thus, the expression \(x^{\frac{5}{4}}\) in radical form is \(\sqrt[4]{x^5}\), which simplifies to \(x\sqrt[4]{x}\) for \(x \ge 0\).
Radical forms are essential in various fields, such as architecture and engineering, where they appear in calculations involving geometry and structures. For instance, if you are designing a square support pillar and need to determine its dimensions based on an area calculation, you might encounter expressions involving roots. Understanding how to convert rational exponents into radicals allows you to interpret and simplify these dimensions correctly, ensuring your structural designs are accurate and safe. For example, if a specific stress calculation results in a load factor of \(L^{\frac{5}{4}}\), expressing it as \(L\sqrt[4]{L}\) can make it much easier to estimate the total load in practical applications.