Simplify the radical expressions in Exercises 67–74, if possible. ³√9⋅³√6

Verified step by step guidance

1. Recognize that the given expression involves the product of two cube roots: ³√9 ⋅ ³√6. Using the property of radicals, ³√a ⋅ ³√b = ³√(a⋅b), combine the two cube roots into a single cube root: ³√(9⋅6).
2. Multiply the numbers inside the cube root: 9 ⋅ 6 = 54. This simplifies the expression to ³√54.
3. Check if the number inside the cube root, 54, can be factored into a perfect cube and another factor. The prime factorization of 54 is 2 ⋅ 3³.
4. Separate the cube root into two parts using the property ³√(a⋅b) = ³√a ⋅ ³√b. Here, ³√54 = ³√(3³) ⋅ ³√2.
5. Simplify the cube root of the perfect cube, ³√(3³), which equals 3. The final simplified expression is 3 ⋅ ³√2.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. They can be simplified by applying properties of exponents and roots, which state that the nth root of a product can be expressed as the product of the nth roots of each factor. Understanding how to manipulate these expressions is crucial for simplification.

Properties of Exponents

The properties of exponents govern how to handle expressions involving powers and roots. For instance, the property that states a^(m/n) = n√(a^m) allows us to convert between radical and exponential forms. This is essential for simplifying radical expressions, as it helps in rewriting them in a more manageable form.

Multiplication of Radicals

When multiplying radical expressions, such as cube roots, the product can be simplified by combining the radicands under a single radical. For example, ³√a ⋅ ³√b = ³√(a*b). This property is key to simplifying expressions like ³√9⋅³√6, as it allows for the multiplication of the numbers inside the radicals before taking the root.