In Exercises 93–104, rationalize each numerator. Simplify, if possible. √x + 4 √x

Verified step by step guidance

1. Identify the expression to rationalize: $\frac{\sqrt{x}+4}{\sqrt{x}}$.
2. Multiply both the numerator and the denominator by the conjugate of the numerator, which is $\sqrt{x}-4$.
3. The expression becomes: $\frac{(\sqrt{x}+4)(\sqrt{x}-4)}{\sqrt{x}(\sqrt{x}-4)}$.
4. Apply the difference of squares formula to the numerator: $(a+b)(a-b)=a^2-b^2$, where $a=\sqrt{x}$ and $b=4$.
5. Simplify the expression by calculating $a^2-b^2$ in the numerator and simplifying the denominator.

Key Concepts

Rationalizing the Numerator

Rationalizing the numerator involves eliminating any square roots or irrational numbers from the numerator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will create a perfect square in the numerator. For example, to rationalize √x + 4, one might multiply by the conjugate, which is √x - 4.

Simplifying Radicals

Simplifying radicals refers to the process of reducing a square root expression to its simplest form. This involves factoring out perfect squares from under the radical sign. For instance, √(x) can be simplified if x is a perfect square, allowing for easier manipulation in algebraic expressions.

Algebraic Fractions

Algebraic fractions are fractions that contain variables in the numerator, denominator, or both. Understanding how to manipulate these fractions, including addition, subtraction, multiplication, and division, is crucial for solving problems involving rational expressions. Simplifying these fractions often requires factoring and canceling common terms.