In Exercises 45–54, rationalize the denominator. 11/(√7−√3)

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically achieved 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 in the form of a binomial with square roots, one can multiply by the conjugate of that binomial.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between two terms. For instance, the conjugate of (a + b) is (a - b). In the context of rationalizing denominators, multiplying by the conjugate helps to eliminate square roots or other irrational components, simplifying the expression into a more manageable form.

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 (√(a*b) = √a * √b) and that the square root of a quotient is the quotient of the square roots (√(a/b) = √a / √b). These properties are crucial when simplifying expressions after rationalizing the denominator.