If the expression is in exponential form, write it in radical form and evaluate if possible. If it is in radical form, write it in exponential form. Assume all variables represent positive real numbers. (5r + 3t)^4/7

Verified step by step guidance

1. Identify the given expression: \((5r + 3t)^{4/7}\). This is in exponential form with a fractional exponent.
2. Recall that a fractional exponent \(a/b\) can be rewritten in radical form as \(\sqrt[b]{x^a}\) or equivalently \(\left(\sqrt[b]{x}\right)^a\).
3. Rewrite the expression \((5r + 3t)^{4/7}\) in radical form as \(\left(\sqrt[7]{5r + 3t}\right)^4\) or \(\sqrt[7]{(5r + 3t)^4}\).
4. Since the problem asks to evaluate if possible, check if the expression inside the radical can be simplified or if numerical values are given. Without specific values, leave the expression in radical form.
5. Thus, the expression in radical form is \(\sqrt[7]{(5r + 3t)^4}\), which is the equivalent radical expression for the given exponential form.

Key Concepts

Exponential and Radical Forms

Exponential form expresses roots and powers using fractional exponents, where the numerator is the power and the denominator is the root. Radical form uses root symbols, such as square roots or cube roots. For example, x^(4/7) in radical form is the 7th root of x raised to the 4th power.

Converting Between Forms

To convert from exponential to radical form, rewrite the fractional exponent as a root and power. Conversely, to convert from radical to exponential form, express the root as a fractional exponent. This conversion helps simplify expressions and solve equations involving powers and roots.

Evaluating Expressions with Positive Variables

When variables represent positive real numbers, fractional exponents and radicals are well-defined and yield real values. This assumption allows safe manipulation of expressions without considering complex numbers or negative roots, simplifying evaluation and ensuring the expression remains valid.