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

Verified step by step guidance

1. Recognize that the symbol \( \sqrt[4]{\cdot} \) represents the fourth root, which means we are looking for a number that, when raised to the power of 4, equals the radicand (the number inside the radical).
2. Express the radicand 243 as a product of its prime factors. Since 243 is a power of 3, write it as \( 243 = 3^5 \).
3. Rewrite the fourth root of 243 using the prime factorization: \( \sqrt[4]{243} = \sqrt[4]{3^5} \).
4. Apply the property of radicals that \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \), so \( \sqrt[4]{3^5} = 3^{\frac{5}{4}} \).
5. Separate the exponent into an integer and fractional part: \( 3^{\frac{5}{4}} = 3^{1 + \frac{1}{4}} = 3^1 \times 3^{\frac{1}{4}} = 3 \times \sqrt[4]{3} \). This is the simplified form of the original radical.

Key Concepts

Understanding Radicals and Roots

A radical expression involves roots, such as square roots or fourth roots. The symbol ∜ represents the fourth root, meaning you are looking for a number that, when raised to the power of 4, equals the given value. Recognizing the type of root is essential for simplification.

Prime Factorization

Prime factorization breaks down a number into its prime factors. For simplifying radicals, expressing the radicand (the number inside the root) as a product of prime factors helps identify perfect powers that can be taken out of the root easily.

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 1/n power. Using exponent rules allows you to simplify radicals by extracting factors raised to powers divisible by the root's index.