Rationalize The Numerator

What Does it Mean To Rationalize The Numerator?

Rationalizing the numerator is the process of eliminating radicals, particularly square roots or cube roots, from the numerator of a fraction. This process is often performed when it's desirable or necessary to have a numerator that is a rational number, or when working with expressions where having the radical in the denominator makes calculation or simplification easier.

Steps to Rationalize The Numerator

To Rationalize The Numerator, there are three steps:

Step 1: Identify the Expression

• Look for a fraction where the numerator has a radical (e.g., $\sqrt{a}$, $\sqrt[3]{a}$) that you want to eliminate.

Step 2: Multiply by a Form of 1

• Multiply both the numerator and denominator by a value that when multiplied with the numerator will remove the radical. This often involves using the conjugate or a similar radical.

Step 3: Simplify

• After multiplication, simplify the fraction if possible. This could involve combining like terms or reducing the fraction.

Key Point:

Rationalizing the numerator is less common than rationalizing the denominator. The primary focus is often on the denominator because mathematical convention prefers to avoid roots in the denominator. However, the principle of multiplying by a form of 1 (like the conjugate) applies to both situations to eliminate radicals without changing the value of the expression.

Four Types of Rationalizing The Numerator

Rationalizing the numerator involves eliminating radicals from the numerator of a fraction. Here are the general methods depending on whether the numerator is a monomial or binomial expression:

1. Rationalizing a Monomial Numerator: For a fraction with a single square root in the numerator, such as $ \frac{\sqrt{a}}{b} $, you would multiply both the numerator and the denominator by the square root that appears in the numerator:

$ \frac{\sqrt{a}}{b} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{a}{b\sqrt{a}} $

The result is a rationalized numerator with the radical now in the denominator.

2. Rationalizing a Binomial Numerator with One Radical: For a numerator that is a binomial with one square root, like $ \frac{\sqrt{a} + c}{b} $, you would multiply the fraction by the conjugate of the numerator over itself to eliminate the radical using the difference of squares:

$ \frac{\sqrt{a} + c}{b} \times \frac{\sqrt{a} - c}{\sqrt{a} - c} = \frac{a - c^2}{b(\sqrt{a} - c)} $

This leaves the numerator in a rationalized form, with any radicals now appearing in the denominator.

3. Rationalizing a Binomial Numerator with Two Radicals: When both terms in the numerator are radicals, such as $ \frac{\sqrt{a} + \sqrt{c}}{b} $, multiply the fraction by the conjugate over itself:

$ \frac{\sqrt{a} + \sqrt{c}}{b} \times \frac{\sqrt{a} - \sqrt{c}}{\sqrt{a} - \sqrt{c}} = \frac{a - c}{b(\sqrt{a} - \sqrt{c})} $

After multiplying, the radical terms in the numerator will be eliminated, with the radicals appearing in the denominator.

4. Rationalizing Numerators with Cube Roots or Higher Roots: For cube roots or other higher roots, the process is similar but may require the use of the conjugate involving the appropriate power:

$ \frac{\sqrt[3]{a}}{b} \times \frac{\sqrt[3]{a^2}}{\sqrt[3]{a^2}} = \frac{a}{b\sqrt[3]{a^2}} $

Each of these methods is aimed at ensuring that the final form of the fraction has no radicals in the numerator, making it simpler for further operations such as addition, subtraction, or integration. It's important to carefully multiply and simplify the resulting expressions after using these formulas.

Solved Examples on Rationalizing the Numerator

Here are five examples of rationalizing the numerator with step-by-step solutions:

Example 1: Rationalizing a Single Radical in the Numerator

Problem: Rationalize the numerator of the fraction $ \frac{\sqrt{5}}{3} $.

Solution 1:

1. Multiply numerator and denominator by $ \sqrt{5} $: $ \frac{\sqrt{5}}{3} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5}{3\sqrt{5}} $

2. The numerator is now rationalized.

Example 2: Rationalizing a Binomial Numerator with One Radical

Problem: Rationalize the numerator of $ \frac{\sqrt{2} + 1}{\sqrt{2} - 3} $.

Solution 2:

1. Identify the conjugate of the numerator, which is $ \sqrt{2} - 1 $.

2. Multiply the fraction by the conjugate over the conjugate: $ \frac{\sqrt{2} + 1}{\sqrt{2} - 3} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{2 - 1}{(\sqrt{2} - 3)(\sqrt{2} - 1)} $

3. Simplify the numerator: $ \frac{1}{(\sqrt{2} - 3)(\sqrt{2} - 1)} $

4. The numerator is now rationalized.

Example 3: Rationalizing a Binomial Numerator with Different Radicals

Problem: Rationalize the numerator of $ \frac{\sqrt{3} + \sqrt{2}}{2} $.

Solution 3:

1. Identify the conjugate of the numerator, which is $ \sqrt{3} - \sqrt{2} $.

2. Multiply the fraction by the conjugate over the conjugate: $ \frac{\sqrt{3} + \sqrt{2}}{2} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{3 - 2}{2(\sqrt{3} - \sqrt{2})} $

3. Simplify the numerator: $ \frac{1}{2(\sqrt{3} - \sqrt{2})} $

4. The numerator is now rationalized.

Example 4: Rationalizing a Numerator with a Cube Root

Problem: Rationalize the numerator of $ \frac{\sqrt[3]{4}}{5} $.

Solution 4:

1. Multiply numerator and denominator by $ \sqrt[3]{4^2} $: $ \frac{\sqrt[3]{4}}{5} \times \frac{\sqrt[3]{4^2}}{\sqrt[3]{4^2}} = \frac{4}{5\sqrt[3]{4^2}} $

2. The numerator is now rationalized.

Example 5: Rationalizing a Complex Numerator

Problem: Rationalize the numerator of $ \frac{2 + i}{3 - i} $.

Solution 5:

1. Identify the conjugate of the numerator, which is $ 2 - i $.

2. Multiply the fraction by the conjugate over the conjugate: $ \frac{2 + i}{3 - i} \times \frac{2 - i}{2 - i} = \frac{4 - 1}{(3 - i)(2 - i)} $

3. Simplify the numerator: $ \frac{3}{(3 - i)(2 - i)} $

4. The numerator is now rationalized.

Rationalizing the numerator is performed to simplify an expression, especially prior to integration or in preparation for further algebraic manipulation. It often involves multiplying by the conjugate, which, in the case of the numerator, transfers the radical to the denominator where it is typically easier to work with.

Five Practices With Answers

Here are several practice problems to help you practice rationalizing the numerator through the above methods. The answers to these example questions have been given. Please complete the steps to solve the problems.

Practice Problem 1: Rationalize the numerator of the fraction $ \frac{\sqrt{3} + 4}{\sqrt{3} - 2} $.

Answer 1: $ \frac{-7}{2 - 4\sqrt{3}} $

Practice Problem 2: Rationalize the numerator of $ \frac{\sqrt{5} - \sqrt{2}}{7} $.

Answer 2: $ \frac{3}{7\sqrt{5} + 7\sqrt{2}} $

Practice Problem 3: Rationalize the numerator of $ \frac{\sqrt[3]{8} + 1}{\sqrt[3]{2}} $.

Answer 3: $ \frac{8}{\sqrt[3]{32}} $

Practice Problem 4: Rationalize the numerator of $ \frac{2\sqrt{7} - 3}{\sqrt{7} + 1} $.

Answer 4: $ \frac{25}{2\sqrt{7} + 1} $

Practice Problem 5: Rationalize the numerator of $ \frac{\sqrt[4]{16} - \sqrt[4]{81}}{5} $.

Answer 5: $ \frac{2 - 3}{5\sqrt[4]{4096}} $

If you encounter some confusion during the exercises, you can also ask for help from SOLVELY WEB (Solvely.Ai), which will provide you with problem-solving steps and explanations.

FAQ About Rationalizing The Numerator

Q: How do you know when to rationalize the denominator or numerator?

A: Rationalizing the denominator or numerator is a process used to eliminate radicals (like square roots) or complex numbers from the bottom (denominator) or top (numerator) of a fraction, aiming for a form that is often considered easier to understand or work with.

When to rationalize the denominator:

• You typically rationalize the denominator when it contains a radical or an imaginary number, and you want your final answer to have only rational numbers or integers in the denominator.
• Example: For $\frac{1}{\sqrt{2}}$, multiply both the numerator and the denominator by $\sqrt{2}$ to get $\frac{\sqrt{2}}{2}$.

When to rationalize the numerator:

• Rationalizing the numerator is less common and usually done for specific purposes, like simplifying an expression for integration or solving an equation. It might be done when the presence of a radical in the numerator complicates further calculations.
• Example: To rationalize the numerator in $\frac{\sqrt{2}}{2}$, you would typically not need to unless you had a specific reason related to the context of a problem.

In summary, rationalizing is done to simplify expressions, especially when a radical or an imaginary number in the denominator complicates addition, subtraction, or further calculations. Rationalizing the numerator is less common and usually context-dependent.

Q: What are the 5 rules for simplifying radicals?

A: The 5 rules for simplifying radicals:

1. Break into prime factors: Decompose the number under the radical into its prime factors. For example, $ \sqrt{50} $ can be broken down into $ \sqrt{2 \times 5 \times 5} $.

2. Pair identical factors: Look for pairs of identical factors. Pairs can be moved outside the radical. Using the previous example, $ \sqrt{2 \times 5 \times 5} $ has a pair of 5's.

3. Move pairs outside radical: Move each pair of identical factors outside the radical as a single factor. From the example, we take the pair of 5's outside to get $ 5\sqrt{2} $.

4. Simplify outside: If there are factors outside the radical, simplify them if possible. In our example, there is nothing to simplify further since we only moved a single pair outside, resulting in $ 5\sqrt{2} $.

5. Combine like terms: If there are similar radical terms, combine them. For instance, $ \sqrt{2} + 3\sqrt{2} = 4\sqrt{2} $.

**Q: How to multiply square roots? **

A: To multiply square roots, follow these steps:

1. Multiply the numbers inside the square roots: $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$.

   Example:

• $\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$.

2. Simplify if possible: If the result under the square root can be simplified (i.e., if it has a perfect square factor), do so.

   Example:

• $\sqrt{4} \times \sqrt{9} = \sqrt{4 \times 9} = \sqrt{36} = 6$.

Remember, simplicity and steps clarity are key.

Q: How do you rationalize the denominator for dummies?

A: To rationalize the denominator:

1. Multiply top and bottom by the conjugate if the denominator is a binomial involving square roots. The conjugate changes the sign between the two terms.

   $\frac{a}{\sqrt{b} + c} = \frac{a}{\sqrt{b} + c} \times \frac{\sqrt{b} - c}{\sqrt{b} - c}$

2. Multiply numerator and denominator by the square root in the denominator if it's a single term.

   $\frac{a}{\sqrt{b}} = \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}$

​ Example 1 (Single term):

​ $\frac{1}{\sqrt{2}} = \frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2}$

​ **Example 2 (Binomial): **

​ $\frac{1}{\sqrt{2} + 1} = \frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{\sqrt{2} - 1}{1}$

Q: What are the rules for rationalizing?

A: Rationalizing involves making a radical expression rational (without radicals in the denominator). Here are the basic rules:

For a Single Square Root in the Denominator:

• Multiply both numerator and denominator by the square root in the denominator.

  Example: $\frac{3}{\sqrt{5}}$ becomes $\frac{3\sqrt{5}}{\sqrt{5}\sqrt{5}} = \frac{3\sqrt{5}}{5}$.

For a Binomial with Square Roots:

• Multiply numerator and denominator by the conjugate of the denominator. The conjugate switches the sign between the two terms.

  Example: $\frac{4}{\sqrt{3} + 1}$ becomes $\frac{4(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{4\sqrt{3} - 4}{2}$.

For Higher Roots:

• If the denominator contains cube roots or fourth roots, multiply by the conjugate or an expression that makes the denominator a perfect cube, fourth power, etc.

  Example (Cube root): $\frac{2}{\sqrt[3]{5}}$ becomes $\frac{2\sqrt[3]{25}}{5}$.

General Tips:

• Aim to eliminate radicals from the denominator.
• The process may involve squaring binomials or utilizing cube roots, depending on the radical.
• Simplify the expression after rationalization if possible.

By following these guidelines, rationalizing the denominator can help make expressions easier to work with, especially when adding, subtracting, or comparing fractions with radical denominators.