In Exercises 65–74, simplify each radical expression and then rationalize the denominator. 25 --------- √5x²y

Verified step by step guidance

1. Identify the expression to simplify: $\frac{25}{\sqrt{5x^{2}y}}$.
2. Simplify the numerator and the denominator separately if possible. Here, the numerator is already simplified.
3. To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt{5x^{2}y}$ to eliminate the square root in the denominator.
4. This results in: $\frac{25\cdot \sqrt{5x^{2}y}}{(\sqrt{5x^{2}y}{)}^2}$.
5. Simplify the expression by calculating $(\sqrt{5x^{2}y}{)}^2=5x^{2}y$ and then simplify the entire fraction.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, the expression √5x²y indicates the square root of the product of 5, x squared, and y. Understanding how to simplify these expressions is crucial, as it involves breaking down the components under the radical to their simplest form.

Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate any radicals from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable radical that will result in a rational number in the denominator. This step is important for presenting the final answer in a standard mathematical form.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by combining like terms, factoring, and eliminating unnecessary components. In the case of radical expressions, this may include simplifying the radical itself and ensuring that any coefficients or variables are expressed in the most concise manner possible, which is essential for clarity and ease of further calculations.