In Exercises 75–92, rationalize each denominator. Simplify, if possible.12------------√7 + √3

Verified step by step guidance

1. Multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{7} - \sqrt{3} \).
2. Write the expression as \( \frac{12(\sqrt{7} - \sqrt{3})}{(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3})} \).
3. Apply the difference of squares formula to the denominator: \((a + b)(a - b) = a^2 - b^2\).
4. Calculate the denominator: \( (\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 \).
5. Simplify the expression to \( \frac{12(\sqrt{7} - \sqrt{3})}{4} \) and then simplify further if possible.

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 sum of square roots, you can multiply by the conjugate of that expression.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between the 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 the expression and is particularly useful in rationalizing denominators that contain square roots.

Simplifying Radicals

Simplifying radicals involves reducing a square root expression to its simplest form. This includes factoring out perfect squares from under the radical sign and rewriting the expression. For example, √12 can be simplified to 2√3, making calculations easier and clearer when working with expressions that include square roots.