In Exercises 1–20, use the product rule to multiply.___ _____⁶√x-5 ⋅ ⁶√(x-5)⁴

Verified step by step guidance

1. Identify the expressions under the sixth root: \( \sqrt[6]{x-5} \) and \( \sqrt[6]{(x-5)^4} \).
2. Apply the product rule for radicals: \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \).
3. Combine the expressions under a single radical: \( \sqrt[6]{(x-5) \cdot (x-5)^4} \).
4. Simplify the expression inside the radical: \( (x-5) \cdot (x-5)^4 = (x-5)^{1+4} = (x-5)^5 \).
5. Express the result as a single radical: \( \sqrt[6]{(x-5)^5} \).

Key Concepts

Product Rule

The product rule is a fundamental principle in algebra that states when multiplying two expressions with the same base, you can add their exponents. For example, a^m * a^n = a^(m+n). This rule is essential for simplifying expressions involving roots and powers, allowing for efficient calculations.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this case, the sixth root (⁶√) indicates that we are dealing with the expression raised to the power of 1/6. Understanding how to manipulate and simplify radical expressions is crucial for solving problems that involve roots.

Exponent Rules

Exponent rules govern how to handle powers and roots in algebra. Key rules include the power of a power (a^(m*n) = a^(mn)) and the power of a product (ab)^n = a^n * b^n. These rules are vital for simplifying expressions that involve both multiplication and roots, ensuring accurate calculations.