In Exercises 50 - 53, rationalize the denominator. 14/(√7 - √5)

Verified step by step guidance

1. Step 1: Recognize that the denominator contains a difference of square roots (√7 - √5). To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator, which is (√7 + √5). This eliminates the square roots in the denominator.
2. Step 2: Write the expression as: (14 / (√7 - √5)) × ((√7 + √5) / (√7 + √5)). This ensures that the value of the fraction remains unchanged because multiplying by the conjugate is equivalent to multiplying by 1.
3. Step 3: Multiply the numerator: 14 × (√7 + √5). This results in 14√7 + 14√5.
4. Step 4: Multiply the denominator using the difference of squares formula: (√7 - √5)(√7 + √5) = (√7)^2 - (√5)^2. Simplify this to 7 - 5, which equals 2.
5. Step 5: Combine the results into a single fraction: (14√7 + 14√5) / 2. Simplify further if necessary by dividing each term in the numerator by 2.

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 a binomial involving 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 its two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which simplifies to a rational number. This technique is essential in rationalizing denominators that contain square roots.

Properties of Square Roots

Understanding the properties of square roots is crucial for manipulating expressions involving them. Key properties include that √a * √b = √(a*b) and that √a / √b = √(a/b). These properties allow for simplification and combination of square root terms, which is often necessary when rationalizing denominators in algebraic expressions.