Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{2}}{\sqrt{7}+2}$$

Short answer

\[\frac{\sqrt{14} - 2\sqrt{2}}{3}\]

Step-by-step solution

Identify the conjugate

To rationalize the denominator, identify the conjugate of the denominator. The conjugate of \(\sqrt{7} + 2\) is \(\sqrt{7} - 2\).

Multiply by the conjugate

Multiply both the numerator and the denominator of the fraction by the conjugate. This gives: \[\frac{\sqrt{2}}{\sqrt{7}+2} \times \frac{\sqrt{7}-2}{\sqrt{7}-2} = \frac{\sqrt{2} (\sqrt{7} - 2)}{(\sqrt{7} + 2)(\sqrt{7} - 2)}\]

Simplify the denominator

Simplify the denominator using the difference of squares formula: \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a=\sqrt{7}\) and \(b=2\). So, the denominator becomes: \[\sqrt{7}^2 - 2^2 = 7 - 4 = 3\]

Simplify the numerator

Distribute \(\sqrt{2}\) in the numerator: \[\sqrt{2} \cdot \sqrt{7} - \sqrt{2} \cdot 2 = \sqrt{14} - 2\sqrt{2}\]

Combine the results

The fraction now is: \[\frac{\sqrt{14} - 2\sqrt{2}}{3}\]

Key concepts

conjugate

When we talk about conjugates in algebra, we refer to a pair of expressions that look similar but have opposite signs in the middle. For instance, the conjugate of \(\sqrt{7} + 2\) is \(\sqrt{7} - 2\).

Why are conjugates useful? They help us eliminate radicals from denominators, making the expressions simpler and easier to work with.

In our given problem, multiplying by the conjugate \(\sqrt{7} - 2\) allows us to change the denominator from a radical expression to a simple numerical value.

difference of squares

The difference of squares is a way to simplify expressions by recognizing that \(a^2 - b^2 = (a + b)(a - b)\). This formula is very useful when dealing with conjugates.

In the problem solution, we used the difference of squares to simplify the denominator:

• Identify \(a = \sqrt{7}\) and \(b = 2\)
• Apply the formula: \(\sqrt{7}^2 - 2^2 = 7 - 4 = 3\)

So, the denominator \(\sqrt{7} + 2\) times its conjugate \(\sqrt{7} - 2\) simplifies to 3, resulting in no radicals.

radicals

Radicals are expressions that involve roots, such as square roots, cube roots, etc. In our problem, we have \(\sqrt{2}\) in the numerator and \(\sqrt{7}\) in the denominator.

Radicals can often make calculations complex, especially in denominators. This is why we aim to 'rationalize the denominator,' i.e., remove the radicals by multiplying by the conjugate. By doing this, calculations become more straightforward and easier to manage.

For example, the radical \(\sqrt{7} + 2\) in the denominator was eliminated by multiplying with \(\sqrt{7} - 2\), leading to no radicals in the final denominator.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. The exercise given involves an algebraic expression with both variables and radicals.

We often need to simplify these expressions to make them more manageable. Here, simplifying involved:

• Identifying and using the conjugate
• Applying the difference of squares
• Distributing the numerator

Ultimately, simplifying the algebraic expression made it possible to present the rationalized form: \(\frac{\sqrt{14} - 2\sqrt{2}}{3}\).