Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 3m / 2 + (√m + n)

Verified step by step guidance

1. Identify the term with the radical in the denominator, which is \( \sqrt{m + n} \).
2. Multiply both the numerator and the denominator of the term by the conjugate of the denominator, which is \( \sqrt{m + n} \).
3. The expression becomes \( \frac{3m}{2} \cdot \frac{\sqrt{m + n}}{\sqrt{m + n}} \).
4. Simplify the expression by multiplying the numerators and the denominators separately.
5. The denominator becomes \( (\sqrt{m + n})^2 = m + n \), and the numerator becomes \( 3m \cdot \sqrt{m + n} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any 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, if the denominator is a square root, multiplying by the same square root can help achieve this.

Properties of Square Roots

Understanding the properties of square roots is essential for manipulating expressions involving them. Key properties include that the square root of a product is the product of the square roots, and the square root of a quotient is the quotient of the square roots. These properties help simplify expressions and are particularly useful when rationalizing denominators.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In the context of rationalizing denominators, recognizing how to combine and simplify these expressions is crucial. This includes understanding how to handle like terms and apply the distributive property when necessary, which is often required when manipulating the numerator and denominator during the rationalization process.