Problem 63 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{4}{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression $\sqrt{\frac{4}{3}}$ Answer: $\frac{2\sqrt{3}}{3}$

Step 1: Identify the numerator and the denominator

The given radical expression is: $$ \sqrt{\frac{4}{3}} $$ The numerator of the fraction is 4, and the denominator is 3.

Step 2: Find the square root of the numerator

Next, we need to find the square root of the numerator: $$ \sqrt{4} = 2 $$

Step 3: Find the square root of the denominator

Now, we find the square root of the denominator: $$ \sqrt{3} $$ Since 3 is a prime number and has no perfect square factors, its square root cannot be further simplified.

Step 4: Compile the simplified radical expression

We found the square root of both the numerator and the denominator. Now, we will compile the simplified radical expression: $$ \frac{\sqrt{4}}{\sqrt{3}} = \frac{2}{\sqrt{3}} $$

Step 5: Rationalize the denominator

Finally, we rationalize the denominator by multiplying both numerator and denominator by the same square root present in the denominator: $$ \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} $$ So, the simplified radical expression is: $$ \frac{2\sqrt{3}}{3} $$

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that include a radical symbol, which is represented as \( \sqrt{} \) for square roots or \( \sqrt[n]{} \) for nth roots. The number under the radical sign is called the radicand. Simplifying radical expressions requires identifying perfect square factors under the radical sign and extracting them.

For example, \( \sqrt{18} \) can be simplified by recognizing that 18 is 9 times 2, and 9 is a perfect square. Therefore, it simplifies to \( 3\sqrt{2} \) because \( \sqrt{9} = 3 \) and the \( \sqrt{2} \) remains under the radical sign.

In the provided exercise, \( \sqrt{\frac{4}{3}} \) is a radical expression with a fractional radicand. Here, the numerator has a perfect square (4), enabling a straightforward simplification.

Rationalizing the Denominator

When a radical, especially a square root, appears in the denominator of a fraction, the process of rationalizing the denominator is used to eliminate it. This is done because having a radical in the denominator is not considered to be in simplest form.

To rationalize the denominator, multiply the fraction by a form of 1 that will eliminate the radical in the denominator. If the denominator is a simple square root, like \( \sqrt{3} \) in our exercise, you would multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \) to get rid of the radical:

\[\frac{\sqrt{4}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\]

This results in a denominator that is a rational number while keeping the value of the fraction unchanged. Rationalizing the denominator is a key step in fully simplifying a radical expression containing a fraction.

Square Root Simplification

Simplifying a square root involves finding the largest perfect square factor of the number under the radical and then taking the square root of that factor. The square root of any perfect square is an integer. Numbers that are not perfect squares will often have square roots that are irrational numbers.

In our example, the square root of 4 is simplified to 2 because 4 is a perfect square, and \( \sqrt{4} = 2 \) is an integer. However, 3 has no such perfect square factor, conserving its radical form as \( \sqrt{3} \) after simplification. Therefore, \( \sqrt{\frac{4}{3}} \) simplifies to \( \frac{2}{\sqrt{3}} \) initially.

Understanding square root simplification is crucial for working with radical expressions, especially when the process involves both numerical and variable radical terms. This concept is foundational for higher levels of algebra and beyond.