Rationalize Denominator: Videos & Practice Problems

To rationalize a denominator with a single radical, multiply both the numerator and denominator by the radical. For example, to simplify $1/\sqrt{3}$, multiply by $\sqrt{3}$ to get $\sqrt{3}/3$. For two-term denominators, use the conjugate to eliminate the radical, ensuring a rational result. This method is essential for maintaining mathematical integrity.

Rationalizing Denominators

In mathematics, it is essential to understand that radicals should not be left in the denominator of a fraction. This practice is known as rationalizing the denominator, which is a straightforward process that allows us to eliminate radicals from the bottom of a fraction. For instance, while expressions like \( \frac{1}{\sqrt{2}} \) can sometimes simplify to perfect squares, others, such as \( \frac{1}{\sqrt{3}} \), cannot be simplified in the same way.

To rationalize the denominator, you multiply both the numerator and the denominator by the radical present in the denominator. For example, if you have \( \frac{1}{\sqrt{3}} \), you would multiply both the top and bottom by \( \sqrt{3} \). This operation is equivalent to multiplying by 1, thus preserving the value of the fraction:

\[\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}\]

In this case, the denominator becomes \( \sqrt{3} \times \sqrt{3} = 3 \), which is a rational number. The resulting expression \( \frac{\sqrt{3}}{3} \) maintains the same value as the original fraction, as confirmed by calculating both expressions, which yield approximately 0.57.

Rationalizing the denominator is a valuable skill in algebra, as it simplifies expressions and makes them easier to work with, especially in further calculations. Remember, while radicals can appear in the numerator, they should always be eliminated from the denominator for clarity and standard mathematical practice.

Rationalize the denominator. 
$-\frac{5}{2\sqrt7}$

• A. $-\frac{5}{14}$
• B. $-\frac{5\sqrt7}{2}$
• C. $-\frac{5\sqrt7}{14}$
• D. $-\frac{10\sqrt7}{14}$

Answer: (C)

Rationalize the denominator. $\frac{6+\sqrt{x}}{-\sqrt{x}}$

• A. $\frac{6\sqrt{x}}{-x}-1$
• B. $6\sqrt{x}+x$
• C. $\frac{6\sqrt{x}}{x}+1$
• D. $\frac{7\sqrt{x}}{-x}$

Answer: (A)

Rationalizing Denominators Using Conjugates

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.

However, complications arise when the denominator consists of two terms, such as 2 + √3. In this case, simply multiplying by the same radical will not suffice, as it leads to a binomial multiplied by a binomial, requiring the use of the FOIL method. This results in a more complex expression that still contains a radical in the denominator, which is undesirable.

To properly rationalize a denominator with two terms, you should use the concept of conjugates. The conjugate of a binomial a + √b is a - √b. By multiplying the fraction by the conjugate of the denominator, you can utilize the difference of squares formula, which states that (a + b)(a - b) = a² - b². This means that when you multiply (2 + √3)(2 - √3), you will get 2² - (√3)² = 4 - 3 = 1, effectively eliminating the radical.

For example, starting with the fraction 1 / (2 + √3), you multiply both the numerator and the denominator by (2 - √3). The result is:

(1 * (2 - √3)) / ((2 + √3)(2 - √3)) = (2 - √3) / 1 = 2 - √3

This process successfully rationalizes the denominator, leaving you with a simplified expression. In summary, when dealing with a single-term denominator, multiply by that term, and when faced with a two-term denominator, multiply by its conjugate to achieve a rational denominator.

Rationalize the denominator and simplify the radical expression.
$\frac{\sqrt7}{5-\sqrt6}$

• A. $\frac{\sqrt7}{19}$
• B. $\frac{5\sqrt7+\sqrt{42}}{-11}$
• C. $\frac{5\sqrt7+\sqrt{42}}{21}$
• D. $\frac{5\sqrt7+\sqrt{42}}{19}$

Answer: (D)

Rationalize the denominator and simplify the radical expression.
$\frac{2-\sqrt3}{2+\sqrt3}$

• A. $7-4\sqrt3$
• B. $\frac{1}{7+4\sqrt3}$
• C. $7+4\sqrt3$
• D. $\frac{1}{7-4\sqrt3}$

Answer: (A)

Here’s what students ask on this topic:

How do you rationalize the denominator of a fraction with a single radical?

To rationalize the denominator of a fraction with a single radical, multiply both the numerator and the denominator by the radical. For example, if you have $\frac{1}{\sqrt{3}}$, you multiply both the top and bottom by $\sqrt{3}$. This gives you $\frac{\sqrt{3}}{3}$. The denominator becomes a rational number because $\sqrt{3}\times \sqrt{3}=3$. So, the fraction simplifies to $\frac{\sqrt{3}}{3}$.

What is the conjugate in rationalizing the denominator?

The conjugate is used to rationalize denominators with two terms, such as $\frac{1}{2+\sqrt{3}}$. The conjugate of $2+\sqrt{3}$ is $2-\sqrt{3}$. Multiply both the numerator and the denominator by this conjugate. This results in $\frac{2-\sqrt{3}}{4-3}$, which simplifies to $\frac{2-\sqrt{3}}{1}$. The denominator becomes rational, eliminating the radical.

Why is it important to rationalize the denominator?

Rationalizing the denominator is important because it simplifies the expression and makes it easier to work with in further calculations. It also adheres to mathematical conventions, which prefer not to have radicals in the denominator. For example, $\frac{1}{\sqrt{3}}$ is less convenient than $\frac{\sqrt{3}}{3}$ for additional operations like addition, subtraction, or comparison of fractions.

How do you rationalize the denominator with two terms?

To rationalize a denominator with two terms, such as $\frac{1}{2+\sqrt{3}}$, use the conjugate. Multiply both the numerator and the denominator by the conjugate of the denominator, which is $2-\sqrt{3}$. This results in $\frac{2-\sqrt{3}}{4-3}$, simplifying to $\frac{2-\sqrt{3}}{1}$. The denominator becomes rational, eliminating the radical.

What happens if you don't rationalize the denominator?

If you don't rationalize the denominator, the expression may be more difficult to work with in further calculations. For example, comparing or adding fractions with radicals in the denominator can be cumbersome. Additionally, it does not adhere to mathematical conventions, which prefer rational denominators for simplicity and clarity. Rationalizing ensures the expression is in its simplest and most usable form.