Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{3}}{5-\sqrt{2}}$$

Short answer

The rationalized form is \( \frac{5 \sqrt{3} + \sqrt{6}}{23} \).

Step-by-step solution

Identify the Conjugate of the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator, which is \(5 - \sqrt{2}\), is \(5 + \sqrt{2}\).

Multiply the Expression by the Conjugate

Multiply the numerator and the denominator by the conjugate: \( \frac{\sqrt{3}}{5 - \sqrt{2}} \times \frac{5 + \sqrt{2}}{5 + \sqrt{2}} = \frac{\sqrt{3} \times (5 + \sqrt{2})}{(5 - \sqrt{2}) \times (5 + \sqrt{2})} \)

Simplify the Numerator

Distribute \(\sqrt{3}\) in the numerator: \(\sqrt{3} \times 5 + \sqrt{3} \times \sqrt{2} = 5 \sqrt{3} + \sqrt{6} \)

Simplify the Denominator

Use the difference of squares formula to simplify the denominator: \( (5 - \sqrt{2}) \times (5 + \sqrt{2}) = 5^2 - (\sqrt{2})^2 = 25 - 2 = 23 \)

Write the Rationalized Expression

Combine the results to obtain the rationalized expression: \( \frac{5 \sqrt{3} + \sqrt{6}}{23} \)

Key concepts

Conjugate of the Denominator

To rationalize a denominator, we often use the conjugate of that denominator. The conjugate changes the sign between two terms. For instance, if the denominator is of the form \(a - b\), its conjugate is \(a + b\). This helps eliminate the square root or irrational part. This is crucial for simplifying expressions and making them easier to work with in further calculations.

Difference of Squares

The difference of squares is a mathematical identity that simplifies expressions involving the subtraction of two square terms. It states that \((a - b)(a + b) = a^2 - b^2\). This identity is particularly useful when rationalizing denominators. In our example, multiplying \((5 - \sqrt{2})(5 + \sqrt{2})\) directly applies this identity, resulting in a simplified expression \(5^2 - (\sqrt{2})^2 = 25 - 2 = 23\). The difference of squares helps to quickly remove the square roots from the denominator.

Simplifying Algebraic Expressions

After using the conjugate, you often end up with a more complex expression in both the numerator and the denominator. The next step involves simplifying these expressions. Simplification generally includes expanding terms like \(\sqrt{3} \times (5 + \sqrt{2})\), resulting in terms like \(5 \sqrt{3} + \sqrt{6}\). Knowing how to distribute and combine like terms is essential for making the expression as simple as possible.

Rational Expressions

Rational expressions consist of a ratio of polynomial expressions. They often contain variables and can include radicals. The goal of rationalizing the denominator is to make these expressions easier to handle. By eliminating the square root from the denominator, we turn a more complex expression into a simpler form such as \( \frac{5 \sqrt{3} + \sqrt{6}}{23} \). This makes it easier for further algebraic manipulation and understanding.

Square Roots

Square roots are the opposite of squaring a number. While square roots can often complicate an expression, rationalizing the denominator replaces the square root in the denominator with simpler terms. In our example, we start with \( \frac{\sqrt{3}}{5 - \sqrt{2}} \). By the end, we obtain \( \frac{5 \sqrt{3} + \sqrt{6}}{23} \). Understanding how to manipulate square roots helps significantly in algebra, especially when simplifying complex fractions.