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

Verified step by step guidance

1. Recognize that the expression is \(-9 \cdot \sqrt[5]{243}\), where \(\sqrt[5]{243}\) is the fifth root of 243.
2. Express 243 as a power of a prime number: since \(243 = 3^5\), rewrite the radical as \(\sqrt[5]{3^5}\).
3. Use the property of radicals and exponents: \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\). So, \(\sqrt[5]{3^5} = 3^{\frac{5}{5}} = 3^1 = 3\).
4. Substitute back into the original expression: \(-9 \cdot 3\).
5. Multiply the constants: \(-9 \times 3\) to get the simplified expression (do not calculate the final value as per instructions).

Key Concepts

Radical Expressions and Simplification

A radical expression involves roots such as square roots or fifth roots. Simplifying radicals means rewriting them in their simplest form by factoring out perfect powers or using properties of exponents. For example, simplifying ⁵√243 involves expressing 243 as a power of a base that matches the root.

Properties of Exponents and Roots

Roots can be expressed as fractional exponents, where the nth root of a number is the same as raising that number to the power of 1/n. This allows the use of exponent rules to simplify expressions, such as rewriting ⁵√243 as 243^(1/5). Understanding this helps in breaking down and simplifying radical expressions.

Handling Negative Coefficients with Radicals

When a negative coefficient multiplies a radical, the negative sign remains outside the root. Since variables represent positive real numbers, the sign does not affect the root's value. For example, -9 ⁵√243 means multiplying -9 by the fifth root of 243, keeping the negative sign separate.