Simplify the radical expressions in Exercises 67–74, if possible. ⁵√64x⁶/⁵√2x

Verified step by step guidance

1. Rewrite the given expression as a single radical: \( \frac{\sqrt[5]{64x^6}}{\sqrt[5]{2x}} = \sqrt[5]{\frac{64x^6}{2x}} \).
2. Simplify the fraction inside the radical: \( \frac{64x^6}{2x} = 32x^5 \).
3. Substitute the simplified fraction back into the radical: \( \sqrt[5]{32x^5} \).
4. Break the radical into two parts: \( \sqrt[5]{32} \cdot \sqrt[5]{x^5} \).
5. Simplify each part: \( \sqrt[5]{32} = 2 \) (since \( 2^5 = 32 \)) and \( \sqrt[5]{x^5} = x \). Combine the results to get \( 2x \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, represented by the radical symbol (√). In this context, we are dealing with fifth roots, denoted as ⁵√. Understanding how to manipulate these expressions, including simplifying them and applying properties of exponents, is crucial for solving problems involving radicals.

Properties of Exponents

The properties of exponents are rules that govern how to handle expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)) and the quotient of powers (a^m / a^n = a^(m-n)). These rules are essential for simplifying expressions with radicals, as they allow us to combine and reduce terms effectively.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. This often includes factoring out perfect powers from under the radical and applying the properties of exponents. For example, when simplifying ⁵√(64x^6), recognizing that 64 is a perfect fifth power (2^6) and x^6 can be expressed as (x^5)(x) helps in reducing the expression.