Perform the indicated operations and/or simplify each expression. Assume all variables represent positive real numbers. 5/√2

Verified step by step guidance

1. Identify the expression that needs simplification: \( \frac{5}{\sqrt{2}} \).
2. Recognize that the expression contains a radical in the denominator, which we should rationalize.
3. Multiply both the numerator and the denominator by \( \sqrt{2} \) to eliminate the radical in the denominator.
4. This results in \( \frac{5 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \).
5. Simplify the denominator using the property \( \sqrt{a} \cdot \sqrt{a} = a \), which gives \( \frac{5\sqrt{2}}{2} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate square roots or other irrational numbers from the denominator of a fraction. This is typically done by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, to rationalize 5/√2, you would multiply by √2/√2, resulting in (5√2)/2.

Properties of Exponents

Understanding the properties of exponents is crucial in algebra, as they dictate how to manipulate expressions involving powers. Key properties include the product of powers, quotient of powers, and power of a power. These rules help simplify expressions and solve equations efficiently, especially when dealing with variables raised to powers.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while maintaining their value. This can include combining like terms, factoring, and rationalizing denominators. The goal is to make expressions easier to work with, which is essential for solving equations or performing further operations.