Simplify each radical. Assume all variables represent positive real numbers. -9 ⁵√243

Verified step by step guidance

1. Identify the expression to simplify: $-9\cdot \sqrt[5]{243}$.
2. Recognize that 243 can be expressed as a power of a smaller number. Notice that $243=3^5$.
3. Substitute $3^5$ for 243 in the expression: $-9\cdot \sqrt[5]{3^5}$.
4. Apply the property of radicals that $\sqrt[n]{a^n}=a$ when $a$ is positive: $\sqrt[5]{3^5}=3$.
5. Multiply the result by $-9$: $-9\cdot 3$.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this context, the expression ⁵√243 indicates the fifth root of 243. Understanding how to simplify radical expressions is essential, as it involves recognizing perfect powers and applying the properties of exponents.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. For instance, when simplifying radicals, one can express roots in terms of fractional exponents, such as ⁵√x being equivalent to x^(1/5). This understanding is crucial for simplifying expressions like -9 ⁵√243.

Perfect Powers

Perfect powers are numbers that can be expressed as an integer raised to an exponent. For example, 243 can be expressed as 3^5, which is a perfect fifth power. Recognizing perfect powers allows for easier simplification of radical expressions, as it enables the extraction of whole numbers from under the radical.