In Exercises 61–82, multiply and simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers.__ _3√15 ⋅ 5√6

Verified step by step guidance

1. Identify the expression to be multiplied: \(3\sqrt{15} \cdot 5\sqrt{6}\).
2. Use the property of radicals: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
3. Multiply the coefficients: \(3 \cdot 5 = 15\).
4. Multiply the radicands: \(\sqrt{15} \cdot \sqrt{6} = \sqrt{15 \cdot 6}\).
5. Simplify the expression: \(15\sqrt{90}\) and further simplify \(\sqrt{90}\) if possible.

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, etc. In this context, the notation √n represents the square root of n, while n√m denotes the nth root of m. Understanding how to manipulate and simplify radical expressions is essential for solving problems involving multiplication and simplification of radicals.

Multiplication of Radicals

When multiplying radicals, the product of the radicands can be taken under a single radical sign. For example, √a ⋅ √b = √(a*b). This property allows for the simplification of expressions by combining the radicands before simplifying further. It is crucial to apply this rule correctly to achieve the simplest form of the expression.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form, which often includes factoring out perfect squares or cubes from the radicand. For instance, √(a*b) can be simplified if a or b is a perfect square. This process is important for presenting the final answer in a clear and concise manner, especially in algebraic expressions.