To "cancel" a square root in a fraction, specifically when it's in the denominator, you usually want to eliminate it through a process called rationalizing the denominator. This involves manipulating the fraction so that the denominator is a rational number (i.e., no square roots).
Rationalizing the Denominator
The most common situation is when you have a square root in the denominator. According to the reference provided, the key is to multiply both the numerator and the denominator by the same root that's in the denominator. This utilizes the principle that multiplying by a fraction equal to 1 (like √2/√2) doesn't change the value of the original fraction, only its form.
Example
Let's say you have the fraction 1/√2.
- Identify the root in the denominator: In this case, it's √2.
- Multiply both numerator and denominator by that root:
(1/√2) * (√2/√2) - Simplify:
- Numerator: 1 * √2 = √2
- Denominator: √2 * √2 = 2
- Result: The fraction becomes √2/2. The denominator is now rationalized.
General Steps
| Step | Description | Example (3/√5) |
|---|---|---|
| 1 | Identify the square root in the denominator. | √5 |
| 2 | Multiply both the numerator and the denominator by that square root. | (3/√5) * (√5/√5) |
| 3 | Simplify the numerator. | 3 * √5 = 3√5 |
| 4 | Simplify the denominator (√x * √x = x). | √5 * √5 = 5 |
| 5 | Write the rationalized fraction. | 3√5 / 5 |
More Complex Denominators
If the denominator is more complex, like a + √b or a - √b, you'll need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate is formed by changing the sign between the two terms in the denominator.
- Example: The conjugate of (2 + √3) is (2 - √3).
By multiplying by the conjugate, you exploit the difference of squares: (a + b)(a - b) = a² - b². This eliminates the square root from the denominator.
Example: Rationalizing 1/(2 + √3)
- Identify the conjugate of the denominator: The conjugate of (2 + √3) is (2 - √3).
- Multiply numerator and denominator by the conjugate:
[1/(2 + √3)] * [(2 - √3)/(2 - √3)] - Simplify:
- Numerator: 1 * (2 - √3) = 2 - √3
- Denominator: (2 + √3)(2 - √3) = 2² - (√3)² = 4 - 3 = 1
- Result: (2 - √3) / 1 = 2 - √3
