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 posi-tive real numbers. (5r + 3t)^4/7

Verified step by step guidance

1. Identify the given expression: \((5r + 3t)^{\frac{4}{7}}\).
2. Recognize that the expression is in exponential form, where the exponent is a fraction.
3. Convert the expression from exponential form to radical form. The expression \((5r + 3t)^{\frac{4}{7}}\) can be rewritten as the 7th root of \((5r + 3t)^4\).
4. Express this in radical form: \(\sqrt[7]{(5r + 3t)^4}\).
5. Note that the expression is now in radical form, and since it involves variables, it cannot be evaluated further without specific values for \(r\) and \(t\).

Key Concepts

Exponential Form

Exponential form represents numbers using a base raised to a power, indicating repeated multiplication. For example, a number 'a' raised to the power 'n' (a^n) signifies that 'a' is multiplied by itself 'n' times. This form is useful for simplifying calculations and expressing large numbers compactly.

Radical Form

Radical form expresses numbers using roots, such as square roots or cube roots. The radical symbol (√) indicates the root of a number, where 'n'√(a) represents the n-th root of 'a'. Converting between radical and exponential forms allows for different perspectives on the same mathematical expression, facilitating easier manipulation and evaluation.

Properties of Exponents and Radicals

Understanding the properties of exponents and radicals is crucial for converting between forms. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the relationship between exponents and roots (a^(1/n) = n√(a)). Mastery of these properties enables effective simplification and evaluation of expressions.