Rationalize each denominator. See Example 8. √3 + 1———— 1 - √3

Verified step by step guidance

1. Identify the expression to rationalize: \( \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \).
2. To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(1 + \sqrt{3}\).
3. Write the expression as: \( \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \).
4. Expand both the numerator and the denominator using the distributive property (FOIL method for binomials).
5. Simplify the expression by combining like terms and using the identity \(a^2 - b^2 = (a-b)(a+b)\) for the denominator.

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 of the form 'a - b', multiplying by 'a + b' can help achieve this.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between its two terms. For instance, the conjugate of 'a - b' is 'a + b'. Using conjugates is a common technique in rationalizing denominators, especially when dealing with square roots, as it simplifies the expression and eliminates the irrational part from the denominator.

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 useful when simplifying expressions after rationalizing the denominator.