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

Verified step by step guidance

1. Step 1: Simplify the radical expression. Start by breaking down the expression under the square root. The expression is \( \sqrt{\frac{150a^3}{b^5}} \).
2. Step 2: Simplify the numerator and the denominator separately. For the numerator, factor 150 as \( 150 = 2 \times 3 \times 5^2 \). For \( a^3 \), it can be written as \( a^2 \times a \).
3. Step 3: Simplify the denominator \( b^5 \) as \( b^4 \times b \).
4. Step 4: Apply the square root to both the numerator and the denominator. Use the property \( \sqrt{xy} = \sqrt{x} \cdot \sqrt{y} \).
5. Step 5: Rationalize the denominator by multiplying the numerator and the denominator by \( b \) to eliminate the square root in the denominator.

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.