Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression by rationalizing the denominator. Simplify your answer.\(\frac{1}{\sqrt{5}}\)

Step-by-Step Solution

Answer

The final, rationalized form of the expression is \(\frac{\sqrt{5}}{5}\).

Step 1: Formulate the Conjugate

Identify that, in this case, the conjugate of the denominator is simply \(\sqrt{5}\) itself. Since there are no other terms, the conjugate is the same as the denominator.

Step 2: Rationalize the Denominator

Multiply both the numerator and denominator by the conjugate. This gives \(\frac{1 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\). It's important to recognize that we're simply multiplying the fraction by a form of 1 (\(\sqrt{5} / \sqrt{5}\)), which doesn't change its value, but does affect its appearance.

Step 3: Simplify the Result

Simplifying the multiplication, we get \(\frac{\sqrt{5}}{5}\). Note that when you multiply \(\sqrt{5}\) by \(\sqrt{5}\), you end up with 5, hence removing the radical from the denominator.

Key Concepts

Conjugate of the Denominator

When encountering a radical in the denominator of a fraction, it's often necessary to rationalize it to make the expression neater and easier to work with. The conjugate of the denominator is a key concept in this process. In the given exercise, the denominator is \(\sqrt{5}\). Since there are no additional terms besides the radical, the conjugate is also \(\sqrt{5}\).

To understand why we use the conjugate, remember that the product of a number and its conjugate results in a rational number. In general, if you have a denominator with a form of \(a + \sqrt{b}\), its conjugate would be \(a - \sqrt{b}\) and vice versa. The idea is to create a product that does not have a radical. In our exercise, we multiply the fraction by \(\sqrt{5}/\sqrt{5}\), which is essentially multiplying by 1, meaning we're not changing the fraction's value, only its form.

Simplifying Radicals

Simplifying radicals is a fundamental concept in algebra that involves rewriting expressions so that the radical sign encompasses the smallest possible number. When we multiply \(\sqrt{5}\) by itself, the result is 5. The square root and the number 5 are inverse operations in this context, thus they 'cancel each other out', and you simply get the number under the radical.

This is grounded in the property that \(\sqrt{a} \cdot \sqrt{a} = a\), which is applied whenever we simplify radicals by multiplication. Simplifying the radical in the given expression, \(\sqrt{5}\), by multiplying it with itself doesn't just follow this property but, as a byproduct, also makes the denominator rational, which was the original objective of the exercise. Hence, the process of simplifying radicals goes hand-in-hand with rationalizing denominators.

Multiplying Fractions

Multiplying fractions is relatively straightforward but it's important not to overlook the procedure when working with more complex elements like radicals. The general rule is to multiply the numerators together and the denominators together. In our exercise, we multiply \(1/\sqrt{5}\) by \(\sqrt{5}/\sqrt{5}\), which results in \(1 \cdot \sqrt{5}\) on the numerator and \(\sqrt{5} \cdot \sqrt{5}\) on the denominator.

The multiplication of the numerators gives us \(\sqrt{5}\), and the multiplication of the denominators results in 5, leading to the simplified fraction of \(\sqrt{5}/5\). Remember that when multiplying fractions, especially when involving radicals, you must always look to simplify whenever possible to achieve the most straightforward expression.