In Exercises 93–104, rationalize each numerator. Simplify, if possible.5√ ---3

Verified step by step guidance

1. Identify the expression to be rationalized: \(\frac{5}{\sqrt{3}}\).
2. Multiply both the numerator and the denominator by the conjugate of the denominator, which in this case is \(\sqrt{3}\), to eliminate the square root from the denominator.
3. Apply the multiplication: \(\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}\).
4. Simplify the denominator using the property of square roots: \(\sqrt{a} \times \sqrt{a} = a\). Thus, \(\sqrt{3} \times \sqrt{3} = 3\).
5. The expression simplifies to \(\frac{5\sqrt{3}}{3}\), which is the rationalized form of the original expression.

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, allowing for simplification.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This process often involves factoring both the numerator and denominator and canceling out any common factors.

Square Roots and Their Properties

Square roots are mathematical expressions that represent a number which, when multiplied by itself, gives the original number. Understanding the properties of square roots, such as √(a*b) = √a * √b and √(a/b) = √a / √b, is essential for manipulating expressions involving square roots effectively.