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Rationalizing the denominator is a common technique when dealing with fractions that have roots in the denominator. It helps to simplify the expression and make it easier to work with. The main idea is to eliminate square roots or other roots from the denominator.

To do this, you multiply both the numerator and the denominator by the conjugate of the denominator if it contains a square root. For example, with the fraction \( \frac{7}{\sqrt{3}} \), we multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \). Since \( \frac{\sqrt{3}}{\sqrt{3}} = 1 \), this operation doesn't change the value of the expression, it just gives it a nicer form.

After multiplication, the fraction becomes \( \frac{7\sqrt{3}}{3} \), removing the square root from the denominator. This is a key step in simplifying radicals, as it allows for easier addition, subtraction, or comparison between fractions.

Adding algebraic fractions often involves finding a common denominator so the fractions can be combined. In the process, like terms are added together to form a single fraction.

In this exercise, after rationalizing the denominator, we had two expressions: \( \frac{7\sqrt{3}}{3} \) and \( 2\sqrt{3} \). To add these, \( 2\sqrt{3} \) needs to be expressed with the same denominator as \( \frac{7\sqrt{3}}{3} \).

### Converting to a Common DenominatorTo accomplish this, express \( 2\sqrt{3} \) as a fraction with 3 as the denominator:

• The expression becomes \( \frac{6\sqrt{3}}{3} \), since 2 times 3 equals 6.
• Now, both terms share the same denominator \( 3 \).

With the same denominator, we can now easily add the fractions:

• Combine \( \frac{7\sqrt{3}}{3} + \frac{6\sqrt{3}}{3} \)
• Yields \( \frac{13\sqrt{3}}{3} \)

This results in a single, simplified fraction.

Combining like terms simplifies expressions by reducing the number of terms into a single term. This is especially useful in both algebraic and radical expressions.

In the given problem, terms involved were \( \sqrt{3} \) values. Following rationalization, we had \( \frac{7\sqrt{3}}{3} \) and \( \frac{6\sqrt{3}}{3} \). Since both terms contain the factor \( \sqrt{3} \), they are like terms.### Adding Like Terms

• You group these like terms together and add their coefficients.

For these expressions:

• Coefficients \( 7 \) and \( 6 \) are added: \( 7 + 6 = 13 \)
• Resulting in \( \frac{13\sqrt{3}}{3} \)

This demonstrates how combining like terms simplifies the problem and concludes in a more concise and manageable form.