Problem 353 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \frac{4}{9 \sqrt{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4 \sqrt{5}}{45} \).

Step 1 - Write the Original Expression

Start with the given expression: \( \frac{4}{9 \, \sqrt{5}} \)

Step 2 - Multiply by a Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{5} \): \( \frac{4}{9 \, \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \)

Step 3 - Simplify the Numerator

Multiply the numerators together: \( 4 \cdot \sqrt{5} = 4 \sqrt{5} \)

Step 4 - Simplify the Denominator

Multiply the denominators together: \( 9 \sqrt{5} \cdot \sqrt{5} = 9 \cdot 5 = 45 \)

Step 5 - Write the Simplified Expression

Combine the results of Steps 3 and 4: \( \frac{4 \sqrt{5}}{45} \)

Step 6 - Simplify Further If Possible

Check if the fraction can be simplified further. In this case, it cannot be simplified further.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a crucial technique when dealing with irrational numbers in fractions. When we rationalize the denominator, we aim to remove the radical (square root or other roots) from the denominator of a fraction. This makes the fraction easier to work with and understand.

For example, consider the expression: \( \frac{4}{9 \sqrt{5}} \). To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{5} \). This makes use of the property that \( \sqrt{a} \cdot \sqrt{a} = a \). So, \( \frac{4}{9 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4 \sqrt{5}} {9 \sqrt{5} \cdot \sqrt{5}} = \frac{4 \sqrt{5}}{45} \).

In this process:

• The numerator changes from 4 to 4\sqrt{5}.
• The denominator becomes 45 since \(9 \sqrt{5} \cdot \sqrt{5}\) simplifies to 9 \cdot 5.

After rationalizing, the entire fraction is often simpler to handle and use in further calculations.

Simplifying Fractions

Simplifying fractions is one of the fundamentals of algebra. It involves reducing the fraction to its simplest form where the greatest common divisor of the numerator and the denominator is 1. In other words, the numerator and the denominator should be as small as possible but still maintain the same value.

For instance, after rationalizing the original expression \( \frac{4}{9 \sqrt{5}} \) to \( \frac{4 \sqrt{5}}{45} \), we should check if there are any common factors that can simplify it further. In this example, 4 and 45 do not have any common factors, thus \( \frac{4 \sqrt{5}}{45} \) is already in its simplest form.

The steps to simplify any fraction are:

• Factor both the numerator and the denominator.
• Find any common factors.
• Divide both the numerator and the denominator by these common factors.

Simplifying fractions helps in keeping calculations straightforward and manageable.

Multiplying Radicals

Multiplying radicals is the process of multiplying expressions that include square roots or other roots. The key concept is to multiply the numbers outside the radical together and then multiply the numbers inside the radicals together.

For example, in the expression \( \frac{4}{9 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \), we multiply the radicals inside to rationalize and simplify: \( 9 \sqrt{5} \cdot \sqrt{5} \). Using the property \( \sqrt{a} \cdot \sqrt{a} = a \), we get \( 9 \cdot 5 = 45 \).

Another example would be multiplying \( \sqrt{3} \cdot \sqrt{12} \). According to the rule, this becomes: \( \sqrt{3 \cdot 12} = \sqrt{36} = 6 \).

Important rules when multiplying radicals:

• Multiply the coefficients (numbers outside the radical signs).
• Multiply the radicands (numbers inside the radical signs).
• Simplify the resultant radical if possible.

Mastering the multiplication of radicals is essential for handling more complex algebraic expressions efficiently.