Question

Rewrite each expression by rationalizing the denominator. $$ \frac{\sqrt{3}}{3 \sqrt{2}+\sqrt{3}} $$

Short answer

Question: Rewrite the following expression by rationalizing the denominator: \(\frac{\sqrt{3}}{3\sqrt{2}+\sqrt{3}}\) Answer: \(\frac{3\sqrt{6} - 3}{15}\)

Step-by-step solution

Identify the conjugate of the denominator

The conjugate of a binomial expression is obtained by changing the sign of the second term. So, the conjugate of the denominator \(3\sqrt{2}+\sqrt{3}\) is \(3\sqrt{2}-\sqrt{3}\).

Multiply the numerator and the denominator by the conjugate

We will multiply both the numerator and the denominator by the conjugate \(3\sqrt{2}-\sqrt{3}\) to eliminate the square roots in the denominator. $$ \frac{\sqrt{3}}{3\sqrt{2}+\sqrt{3}} \times \frac{3\sqrt{2}-\sqrt{3}}{3\sqrt{2}-\sqrt{3}} $$

Calculate the product of the numerators and denominators

Now, multiply the numerators together and the denominators together, and simplify. $$ \frac{\sqrt{3}(3\sqrt{2}-\sqrt{3})}{(3\sqrt{2}+\sqrt{3})(3\sqrt{2}-\sqrt{3})} $$

Simplify the numerator

Distribute the \(\sqrt{3}\) in the numerator: $$ \frac{3\sqrt{6} - 3}{(3\sqrt{2}+\sqrt{3})(3\sqrt{2}-\sqrt{3})} $$

Simplify the denominator

Apply the difference of squares (a^2 - b^2 = (a+b)(a-b)) formula to simplify the denominator: $$ \frac{3\sqrt{6} - 3}{(3\sqrt{2})^2 - (\sqrt{3})^2} $$ $$ \frac{3\sqrt{6} - 3}{18 - 3} $$

Final simplification

Simplify the denominator and the expression: $$ \frac{3\sqrt{6} - 3}{15} $$

Key concepts

Conjugate

When we talk about the conjugate in mathematics, especially when dealing with expressions involving radicals, we are referring to a technique used to eliminate square roots from the denominator. This process involves using a binomial expression.

• To form the conjugate of a binomial, simply change the sign between the two terms. For instance, if you have a binomial like \( 3\sqrt{2} + \sqrt{3} \), the conjugate would be \( 3\sqrt{2} - \sqrt{3} \).

The beauty of using the conjugate is that when you multiply a binomial by its conjugate, the middle terms cancel out. This results in a simpler expression, often utilizing something called the difference of squares, which makes the expression cleaner. This method becomes especially handy when you're trying to rationalize a denominator that involves a binomial with a square root.

Difference of Squares

The difference of squares is a helpful algebraic identity that states \( a^2 - b^2 = (a+b)(a-b) \). This equation is fundamental when working with rationalizing denominators using conjugates. Here's how it works:

• When you multiply two conjugates, such as \((a+b)(a-b)\), you result in the difference of squares: \(a^2 - b^2\).
• This effectively removes any terms involving \(ab\).

In the context of our problem, the conjugate multiplication \((3\sqrt{2}+\sqrt{3})(3\sqrt{2}-\sqrt{3})\) simplifies to \((3\sqrt{2})^2 - (\sqrt{3})^2\), which is \(18 - 3\). This directly employs the difference of squares to simplify the denominator from a radical to a whole number. Using this identity is crucial because it allows us to clearly and effectively simplify complex expressions.

Simplification

Simplification in mathematics refers to the process of reducing an expression to its simplest form. This usually means rewriting the expression without any unnecessary complexity, such as eliminating radicals in the denominator. It plays a crucial role in making equations more manageable.

• In our exercise, we simplified the expression by using the conjugate of the denominator to multiply both numerator and denominator, leading to easy simplification of radicals.
• This process typically involves applying algebraic identities, such as the difference of squares, to make the expression simpler and reduce forms like square roots in the denominator. Ultimately, the goal is to ensure the expression is as clear as possible, making it easier to work with or to understand.

The final simplification steps allowed us to cleanly present the expression as \((3\sqrt{6} - 3)/15\), which had no radicals in the denominator.

Binomials

A binomial is a polynomial expression with two terms. Understanding how binomials work is fundamental when dealing with rationalizing the denominator.

• They can take the form of \(ax + b\) or \(a+b\), where \(a\) and \(b\) are terms.
• Binomials are important because they often appear in expressions we need to simplify. This is particularly true when they include radicals.

In our example, the denominator \(3\sqrt{2} + \sqrt{3}\) is a binomial involving radicals. The process of rationalizing a denominator frequently involves transforming this binomial by using its conjugate and then simplifying accordingly. Recognizing how binomials function is key as it allows us to systematically approach and simplify expressions by strategically using algebraic identities.