Rationalizing the denominator is an important technique in algebra, particularly when dealing with fractions that contain radicals. When you encounter a fraction like 1 over √3, you can eliminate the radical by multiplying both the numerator and the denominator by √3. This process effectively removes the radical from the denominator, resulting in a simplified expression.

However, when the denominator consists of two terms, such as 2 + √3, simply multiplying by the same radical will not suffice. In this case, you need to use the concept of conjugates. The conjugate of a binomial is formed by changing the sign between the two terms. For example, the conjugate of 2 + √3 is 2 - √3.

To rationalize a denominator with two terms, multiply both the numerator and the denominator by the conjugate. This means that for the expression 1 over 2 + √3, you would multiply by 2 - √3. The multiplication in the denominator will yield a difference of squares:

\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

In the numerator, you multiply straight across:

\[ 1 \cdot (2 - \sqrt{3}) = 2 - \sqrt{3} \]

Thus, the expression simplifies to:

\[ \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3} \]

This process effectively rationalizes the denominator, resulting in a rational number. Remember, when dealing with a single-term denominator, multiply by that term, and when faced with a two-term denominator, always use the conjugate to eliminate the radical. This technique is essential for simplifying expressions and solving equations in algebra.

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