Problem 348 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \frac{4}{\sqrt{5}} $$

Step-by-Step Solution

Answer

\( \frac{4\sqrt{5}}{5} \)

Step 1 - Identify the need for rationalization

Notice that the denominator contains a square root, which must be rationalized.

Step 2 - Multiply by the conjugate

To rationalize, multiply both numerator and denominator by the conjugate of the denominator, which is \( \sqrt{5} \) in this case: \( \frac{4}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\ = \frac{4 \sqrt{5}}{5}\ \).

Step 3 - Simplify the fraction

Simplify the resulting expression if needed. Our fraction is already simplified: \( \frac{4\sqrt{5}}{5} \).

Key Concepts

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 multiplied by 5 equals 25. You denote square roots using the radical symbol √.

When dealing with fractions, finding the square root of a number in the denominator can be tricky. Suppose you have the fraction \(\frac{4}{\rightarrow \sqrt{5}}\), and you need to rationalize it. You need to get rid of the square root in the denominator.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form.

Here are some steps to simplify fractions:

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.
• If the numerator and denominator no longer share any common factors other than 1, the fraction is simplified.

Let's look at our problem: \(\frac{4}{\rightarrow \sqrt{5}}\). Here, the fraction needs rationalization before simplification. This brings us to the next concept—rationalizing using conjugates!

Conjugates

Conjugates are pairs of expressions involving square roots or complex numbers that, when multiplied together, turn into rational numbers. For square roots, the conjugate of \(\rightarrow \sqrt{a}\) is \(\rightarrow \sqrt{a}\) itself.

To rationalize \(\frac{4}{\rightarrow \sqrt{5}}\), multiply the fraction by the conjugate of the denominator:

\(\frac{\rightarrow4}{\rightarrow \sqrt{5}}\) \(\rightarrow \times \rightarrow \frac{\rightarrow \sqrt{5}}{\rightarrow \sqrt{5}}\)

This result simplifies to \(\rightarrow \frac{\rightarrow4 \rightarrow \sqrt{5}}{5}\). The denominator is now rationalized, and that's how we arrive at our final answer.

Using conjugates makes it easier to handle these expressions and simplify fractions with square roots in the denominator.