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.__ ___√8x ⋅ √10y

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1. Recognize that you are multiplying two square roots: \( \sqrt{8x} \) and \( \sqrt{10y} \).
2. Use the property of square roots that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) to combine the square roots: \( \sqrt{8x} \cdot \sqrt{10y} = \sqrt{8x \cdot 10y} \).
3. Multiply the expressions inside the square root: \( 8x \cdot 10y = 80xy \).
4. Simplify the square root \( \sqrt{80xy} \) by factoring 80 into its prime factors: \( 80 = 2^4 \cdot 5 \).
5. Rewrite \( \sqrt{80xy} \) as \( \sqrt{2^4 \cdot 5 \cdot xy} \) and simplify by taking the square root of the perfect square factor \( 2^4 \).

Key Concepts

Properties of Square Roots

The properties of square roots state that the square root of a product is equal to the product of the square roots. This means that √a ⋅ √b = √(a ⋅ b). This property is essential for simplifying expressions involving square roots, allowing us to combine terms under a single radical.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the square root to its simplest form. This includes factoring out perfect squares from the radicand. For example, √(4x) can be simplified to 2√x, making it easier to work with in calculations.

Multiplication of Variables

When multiplying variables, it is important to combine like terms and apply the laws of exponents. For instance, when multiplying x^m by x^n, the result is x^(m+n). This concept is crucial when dealing with expressions that include variables under square roots, ensuring accurate simplification.