What is how to rationalize the denominator?

Rationalizing the denominator is a process used to eliminate radicals (like square roots, cube roots, etc.) from the denominator of a fraction. This makes the fraction simpler to work with in many cases. Here's a breakdown of how to do it:

When the Denominator Contains a Single Radical Term:

• Identify the radical in the denominator.
• Multiply both the numerator and the denominator of the fraction by that radical. This is effectively multiplying by 1, so the value of the fraction doesn't change.
• Simplify the resulting expression. The radical in the denominator should now be gone.

Example:

To rationalize 1/√2, you would multiply both numerator and denominator by √2:

(1/√2) * (√2/√2) = √2/2

When the Denominator Contains a Binomial with a Radical Term:

This usually involves a sum or difference of two terms, where at least one term contains a radical.

• Find the conjugate of the denominator. The conjugate is formed by changing the sign between the two terms. For example, the conjugate of a + √b is a - √b, and the conjugate of √a - √b is √a + √b.
• Multiply both the numerator and the denominator of the fraction by the conjugate.
• Simplify the expression. Use the difference of squares formula: (a + b)(a - b) = a² - b² . This will eliminate the radical in the denominator.

Example:

To rationalize 1/(1 + √3), you would multiply both numerator and denominator by the conjugate (1 - √3):

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

Key Concepts and Tips:

• <a href="https://www.wikiwhat.page/kavramlar/Radicals">Radicals</a>: Understanding what a radical is (square root, cube root, etc.) and how to simplify them is essential.
• <a href="https://www.wikiwhat.page/kavramlar/Conjugate%20(Mathematics)">Conjugate (Mathematics)</a>: Knowing how to find the conjugate of a binomial expression is crucial for rationalizing denominators of that form.
• <a href="https://www.wikiwhat.page/kavramlar/Difference%20of%20Squares">Difference of Squares</a>: The algebraic identity (a + b)(a - b) = a² - b² is very important when rationalizing denominators containing binomials with radicals.
• Simplification: After multiplying by the radical or conjugate, always simplify the resulting fraction as much as possible. Look for common factors in the numerator and denominator.

By following these steps, you can effectively rationalize the denominator of a fraction and simplify your mathematical expressions.