Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. $\sqrt{10x}\cdot \sqrt{8x}$

Verified step by step guidance

1. Identify the two functions being multiplied: here, the expressions are \( \sqrt{10x} \) and \( \sqrt{8x} \).
2. Recall the product rule for square roots: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Use this to combine the two square roots into one.
3. Multiply the expressions inside the square roots: \( 10x \times 8x = 80x^2 \). So, the expression becomes \( \sqrt{80x^2} \).
4. Simplify the square root by factoring out perfect squares. For example, \( 80 = 16 \times 5 \), and \( x^2 \) is a perfect square.
5. Rewrite the expression as \( \sqrt{16 \times 5 \times x^2} = \sqrt{16} \times \sqrt{5} \times \sqrt{x^2} \), then simplify each square root separately.

Key Concepts

Product Rule for Radicals

The product rule for radicals states that the square root of a product equals the product of the square roots: √a * √b = √(a*b). This rule allows simplification by combining radicals under a single root when multiplying.

Properties of Exponents and Radicals

Radicals can be expressed as fractional exponents, such as √x = x^(1/2). Understanding this helps in manipulating and simplifying expressions involving roots and powers, especially when variables are involved.

Domain Restrictions for Variables

When variables are under square roots, they must be nonnegative to keep the expression real. This restriction ensures the expression is defined within the real numbers and avoids complex results.