二次根式化简求值约分法

二次根式化简求值约分法
英文回答：
Simplifying Radicals Using Rationalization.
Simplifying radicals, also known as rationalizing denominators, is a mathematical technique employed to eliminate irrationality from the denominator of a fraction by multiplying both the numerator and denominator by a suitable expression. This process results in a simplified fraction with a rational denominator, making it easier to perform further calculations and simplify algebraic expressions.
The fundamental principle behind rationalization is to transform the denominator into a form that can be easily factored as the product of rational numbers. To achieve this, we identify the term that contains the radical sign in the denominator and multiply both the numerator and denominator by its rational conjugate, which is essentially
the same term but with the opposite sign between the
radical and the rational number underneath.
This technique works because the product of a number and its conjugate always produces a rational result. By applying this principle, we can effectively rationalize the denominator, eliminating the irrationality and leaving us with a fraction that can be easily simplified.
Step-by-Step Process:
1. Identify the Radical in the Denominator: Locate the term in the denominator that contains the radical sign.
2. Find the Rational Conjugate: Multiply the radical term by itself, but change the sign between the radical and the rational number underneath.
3. Multiply Numerator and Denominator: Multiply both the numerator and denominator of the fraction by the rational conjugate.
4. Simplify the Product: Expand and simplify the product to eliminate the radical from the denominator.
5. Reduce the Fraction: If possible, simplify the resulting fraction by reducing it to its lowest terms.
Example:
Simplify the following radical expression:
1 / (√5 2)。

Solution:
1. Identify the radical in the denominator: √5 2。

2. Find the rational conjugate: √5 + 2。

3. Multiply numerator and denominator:
(1 / (√5 2)) ((√5 + 2) / (√5 + 2))。

4. Simplify the product:
= (√5 + 2) / ((√5 2) (√5 + 2))。

= (√5 + 2) / (5 4)。

= (√5 + 2) / 1。

5. Reduce the fraction:
= √5 + 2。

Therefore, the simplified expression is √5 + 2.
中文回答：
二次根式化简求值约分法。

二次根式化简，又称有理化分母，是一种数学技巧，通过将分子和分母同时乘以一个合适的表达式来消除分母中的无理数。

这个过程产生了一个分母有理化的简化分数，便于进行进一步的计算和代数表达式的简化。

有理化背后的基本原理是将分母转换成一个可以很容易地分解
为有理数乘积的形式。

为了实现这一点，我们找出分母中包含根号
的项，并将分子和分母同时乘以其有理共轭，它本质上是相同的项，但在根号和下面的有理数之间带有相反的符号。

这种技巧之所以奏效，是因为一个数与其共轭的乘积总是产生
有理结果。

通过应用这个原理，我们可以有效地有理化分母，消除
无理数，并得到一个可以很容易简化的分数。

分步流程：
1. 找出分母中的根号，找到分母中包含根号的项。

2. 求出有理共轭，将根号项乘以它本身，但改变根号和根号之
下的有理数之间的符号。

3. 分子分母同时乘，将分数的分子和分母同时乘以有理共轭。

4. 简化乘积，展开并简化乘积以消除分母中的根号。

5. 约分，如果可能，通过约分将结果分数简化为最简分数。

例题：
化简下列根式表达式：
1 / (√5 2)。

解题：
1. 分母中找出根号，√5 2。

2. 求出有理共轭，√5 + 2。

3. 分子分母同时乘：
(1 / (√5 2)) ((√5 + 2) / (√5 + 2))。

4. 简化乘积：
= (√5 + 2) / ((√5 2) (√5 + 2))。

= (√5 + 2) / (5 4)。

= (√5 + 2) / 1。

5. 约分：
= √5 + 2。

因此，简化后的表达式为√5 + 2。