Problem 77 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \frac{\sqrt{3}}{6+\sqrt{6}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $\frac{6\sqrt{3} - 3\sqrt{6}}{30}$.

Step 1: Identify the conjugate of the denominator

The conjugate of a term is formed by changing the sign of the term with the square root. The conjugate of the denominator (6 + sqrt(6)) would be (6 - sqrt(6)).

Step 2: Rationalize the denominator

To rationalize the denominator, we'll multiply the numerator and denominator by the conjugate of the denominator, without changing the value of the expression. This is because multiplying and dividing by the same number is equivalent to multiplying by 1. The expression becomes: $$ \frac{\sqrt{3}}{6+\sqrt{6}} \times \frac{6-\sqrt{6}}{6-\sqrt{6}} $$

Step 3: Perform the multiplication

Now, multiply the numerators and the denominators separately. Numerator: $$(\sqrt{3})(6-\sqrt{6}) = 6\sqrt{3} - 3\sqrt{6}$$ Denominator: $$ (6 + \sqrt{6})(6 - \sqrt{6}) = 6^2 - (\sqrt{6})^2 = 36 - 6 = 30$$ Now, substitute the results: $$ \frac{6\sqrt{3} - 3\sqrt{6}}{30} $$

Step 4: Simplify the expression, if possible

Check the expression for any common factors that could simplify the fraction. In this case, there doesn't seem to be any common factor for all terms, so the expression is simplified. The final simplified expression is: $$ \frac{6\sqrt{3} - 3\sqrt{6}}{30} $$

Key Concepts

Rationalizing the Denominator

When it comes to simplifying radical expressions, one crucial skill is rationalizing the denominator. This involves transforming the expression so that the denominator is a rational number, i.e., a number without a radical (square roots, cube roots, etc.). To achieve this, a common method is to multiply the expression by a form of 1 that eliminates the radical in the denominator. This 'form of 1' is often the conjugate of the denominator when it contains a radical.

In our example, we multiply the given expression by the conjugate of the denominator, ensuring that the overall value of the expression remains unchanged. This multiplication leads to a difference of squares in the denominator, turning the radical into a rational number. Afterwards, we may even find some common factors to simplify further. When rationalizing the denominator, take special care to multiply both the numerator and the denominator by the conjugate to keep the original value of the expression intact.

Conjugates in Algebra

The concept of conjugates is pivotal when working with complex numbers and radical expressions in algebra. A conjugate in algebra features a pair of binomials that are identical except for the opposite signs between their terms. For instance, if we have a binomial like \(a + b\), the conjugate would be \(a - b\). This property is particularly handy when you're trying to eliminate radicals from the denominator of a fraction.

By multiplying the fraction by the conjugate over itself, we leverage the difference of squares pattern, which results in a radical-free denominator. The critical point to remember is that conjugates always produce a difference of squares when multiplied together, which simplifies the radical terms. Knowing when and how to use conjugates will significantly advance your ability to simplify and understand radical expressions.

Multiplying Radicals

Multiplying radicals, that is, expressions containing square roots or higher roots, follows specific arithmetic rules. When the radicals have the same index (like square roots), you can multiply the radicands (the numbers inside the radical sign) together while keeping them under a single radical. For instance, \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).

In the context of our exercise, multiplying radicals comes into play when we multiply the numerator. Handling this multiplication carefully is important because it allows us to keep the equation intact and obtain the terms of the simplified numerator. This is otherwise a straightforward process, but it can become tricky if not approached methodically. Always look for opportunities to simplify the radicands or even factor them out to make the expression neater and more comprehensible.