Question

Rationalize the denominator of \(\frac{3}{\sqrt{7}-2}\)

Short answer

\root{7} + 2.

Step-by-step solution

Identify the Conjugate

To rationalize the denominator of \(\frac{3}{\root{7} - 2}\), identify the conjugate of the denominator, which is \(\root{7} + 2\). Using the conjugate will help to eliminate the square root in the denominator.

Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate, \(\root{7} + 2\). This gives us \(\frac{3 (\root{7} + 2)}{(\root{7} - 2)(\root{7} + 2)}\).

Simplify the Denominator

Use the difference of squares formula \((a - b)(a + b) = a^2 - b^2\) to simplify the denominator: \((\root{7})^2 - 2^2 = 7 - 4 = 3\). Thus, the expression becomes \(\frac{3 (\root{7} + 2)}{3}\).

Cancel Common Factors

Cancel the common factor of 3 in both the numerator and denominator: \(\frac{3 (\root{7} + 2)}{3} = \root{7} + 2\).

Key concepts

Conjugate

When rationalizing a denominator, one powerful technique is to use the conjugate. The conjugate of a binomial expression is created by changing the sign between its terms. For instance, the conjugate of \(\sqrt{7} - 2\) is \(\sqrt{7} + 2\). By multiplying both the numerator and denominator by this conjugate, we can eliminate the radical in the denominator.
This step makes further simplification easier and ensures the expression has a rational denominator, which is often a desired form in math.

Difference of Squares

Applying the conjugate takes advantage of the difference of squares formula, which states \((a - b)(a + b) = a^2 - b^2\). When you multiply \(\sqrt{7} - 2\) by its conjugate, \(\sqrt{7} + 2\), the denominator becomes \(\sqrt{7}^2 - 2^2\). This simplifies to \(7 - 4 = 3\).
The difference of squares formula is vital because it transforms an expression with radicals into a simpler, rational form, paving the way for further reduction and simplification.

Cancel Common Factors

After simplifying the denominator using the difference of squares, check if there are any common factors in the numerator and denominator that can be canceled. In our example, \(\frac{3(\sqrt{7} + 2)}{3}\), the 3 in the numerator and denominator can be canceled to yield \(\sqrt{7} + 2\).
Cancelling common factors helps to reduce the expression to its simplest form, making it cleaner and easier to interpret.

Simplifying Radicals

Simplifying radicals involves expressing the square root in the simplest form. For example, \(\sqrt{7}\) is already simplified as it cannot be broken down any further without resulting in decimals or fractions.
Throughout this process, aiming to keep radicals as simplified as possible ensures that expressions are easy to handle and work with. By using conjugates and differences of squares, we effectively manage and simplify radicals in more complex expressions, resulting in clean, rational forms.