In Exercises 65–74, simplify each radical expression and then rationalize the denominator. 150a³ - √ ---------- b⁵

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and can be simplified by factoring out perfect squares or cubes. Understanding how to manipulate these expressions is crucial for simplifying them effectively. For example, √(a²) simplifies to 'a', which is a fundamental property of radicals.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any radicals from the denominator of a fraction. This is typically done by multiplying both the numerator and the denominator by a suitable radical that will result in a rational number in the denominator. For instance, to rationalize 1/√b, you would multiply by √b/√b.

Properties of Exponents

Properties of exponents are rules that govern how to handle expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)) and the power of a power ( (a^m)^n = a^(m*n)). These properties are essential for simplifying expressions that contain variables raised to powers, especially when combined with radicals.