In Exercises 45–66, divide and, if possible, simplify. ____ √50xy 2√2

Verified step by step guidance

1. Step 1: Start by simplifying the expression under the square root in the numerator, $\sqrt{50xy}$.
2. Step 2: Recognize that $50=25\times 2$, so $\sqrt{50xy}=\sqrt{25\times 2\times x\times y}$.
3. Step 3: Simplify $\sqrt{25\times 2\times x\times y}$ to $\sqrt{25}\times \sqrt{2}\times \sqrt{x}\times \sqrt{y}=5\sqrt{2xy}$.
4. Step 4: Divide the simplified numerator $5\sqrt{2xy}$ by the denominator $2\sqrt{2}$.
5. Step 5: Simplify the division by canceling out common terms, $\frac{5\sqrt{2xy}}{2\sqrt{2}}=\frac{5\sqrt{xy}}{2}$.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, √50xy represents the square root of the product of 50, x, and y. Understanding how to manipulate these expressions, including simplifying them by factoring out perfect squares, is essential for solving the problem.

Simplifying Radicals

Simplifying radicals involves rewriting a radical expression in its simplest form. This includes breaking down the radicand (the number or expression inside the root) into its prime factors and identifying perfect squares. For example, √50 can be simplified to 5√2, which is crucial for further calculations in the division.

Dividing Radicals

Dividing radicals requires applying the property that states √a / √b = √(a/b). This means that when dividing two radical expressions, you can combine them under a single radical. In this exercise, dividing √50xy by 2√2 involves simplifying the expression under the radical and ensuring that the final answer is in its simplest form.