Problem 358 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \sqrt{\frac{4}{27}} $$

Step-by-Step Solution

Answer

\(\frac{2\text{\text{ square root of 3 }}}{9}\underline{\phantom{xxx}} \).

Step 1: Simplify the fraction inside the square root

Simplify the fraction \(\frac{4}{27}\). In this case, it is already in its simplest form.

Step 2: Separate the square root of the numerator and the denominator

Rewrite \(\frac{4}{27}\) inside the square root as \(\frac{\text{numerator}}{\text{denominator}}\): \(\frac{\text{numerator}}{\text{denominator}} = \frac{\text{4}}{\text{27}}\).

Step 3: Apply the square root to the numerator and the denominator

Use the property \(\frac{\text{a}}{\text{b}} = \frac{\text{\text{square root of a}}}{\text{\text{square root of b}}}\) inside the square root: \(\frac{\text{\text{square root of 4}}}{\text{\text{square root of 27}} = \frac{\text{2}}{\text{\text{3 square root of 3}}}}\).

Step 4: Rationalize the denominator

Multiply the numerator and the denominator by \(\text{square root of 3}\) to eliminate the square root from the denominator: \(\frac{2}{3 \text{\text{ square root of 3 }}} \times \frac{\text{\text{square root of 3}}}{\text{\text{square root of 3}}} = \frac{2 \text{\text{square root of 3}}}{9 \).

Key Concepts

simplifying fractions

Simplifying fractions means reducing them to their lowest terms. This involves dividing both the numerator and the denominator by their greatest common divisor.
For example, if you have a fraction like \(\frac{4}{8}\), it can be simplified to \(\frac{1}{2}\) by dividing both the top and bottom by 4, which is their greatest common divisor.

In this particular exercise, \(\frac{4}{27}\) is already in its simplest form, so there is no further reduction needed.
Knowing when a fraction is already simplified is a key skill in math as it makes further calculations much easier.

square root property

The square root property is very useful when dealing with fractions inside a square root. You can separate the square root of a fraction into the square root of the numerator and the square root of the denominator. Mathematically, this is represented as:
\[ \frac{\text{square root of a}}{\text{square root of b}} = \text{square root of} \frac{a}{b} \]
For example, if you have \[ \text{\text{the square root of}} \frac{4}{27} \], you can break it down into:
\[ \frac{\text{square root of 4}}{\text{square root of 27}} \]
This helps you to deal with simpler numbers, making further steps clearer. In our case, the numerator becomes 2 (since the square root of 4 is 2) and the denominator remains more complex at \[ 3 \text{ square root of 3} \].

rationalizing denominators

Rationalizing the denominator involves removing any square roots from the denominator of a fraction.
This is done by multiplying both the numerator and the denominator by a number that will eliminate the square root in the denominator.
For instance, if you have \[ \frac{2}{3 \text{ square root of 3}} \], you can multiply both parts by \[ \text{ square root of 3} \] to get:
\[ \frac{2 \text{ square root of 3}}{3 \text{ square root of 3} \times \text {square root of 3} } = \frac{2 \text{ square root of 3}}{9} \]
This makes the fraction easier to work with and more standardized.
Rationalizing the denominator is often the last step in simplifying expressions and is important to master for clearer mathematical communication.