Simplify each radical. Assume all variables represent positive real numbers. - ∜243

Verified step by step guidance

1. Identify the expression to simplify: \(-\sqrt[4]{243}\).
2. Recognize that 243 can be expressed as a power of a smaller number. Note that 243 = 3^5.
3. Rewrite the expression using the property of radicals: \(-\sqrt[4]{3^5}\).
4. Apply the property of radicals: \(\sqrt[n]{a^m} = a^{m/n}\). In this case, \(3^{5/4}\).
5. Simplify the expression: \(-3^{5/4}\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The notation ∜243 indicates the fourth root of 243. Understanding how to simplify these expressions is crucial, as it involves recognizing perfect powers and applying the properties of exponents.

Properties of Exponents

The properties of exponents govern how to manipulate expressions involving powers and roots. For instance, the nth root of a number can be expressed as that number raised to the power of 1/n. This concept is essential for simplifying radical expressions, as it allows for the conversion between radical and exponential forms.

Perfect Powers

Perfect powers are numbers that can be expressed as an integer raised to an exponent. For example, 243 is a perfect power since it can be expressed as 3^5. Recognizing perfect powers is key to simplifying radicals, as it allows for the extraction of whole numbers from under the radical sign.