---
title: Simplifying Square Roots
localeTitle: 简化平方根
---
### 示例：简化![](https://latex.codecogs.com/gif.latex?\sqrt{180} "\ SQRT {180}")

解决步骤：

*   将180分为素数因子的乘积： ![](https://latex.codecogs.com/gif.latex?180&space;=&space;2^2&space;*&space;3^2&space;*&space;5 "180 = 2 ^ 2 * 3 ^ 2 * 5")
*   使用激进的产品属性![](https://latex.codecogs.com/gif.latex?\sqrt{ab}&space;=&space;\sqrt{a}\sqrt{b}&space;\Rightarrow&space;\sqrt{180}&space;=&space;\sqrt{2^2}\sqrt{3^2}\sqrt{5}=2*3\sqrt{5}=6\sqrt{5} "\ sqrt {ab} = \ sqrt {a} \ sqrt {b} \ Rightarrow \ sqrt {180} = \ sqrt {2 ^ 2} \ sqrt {3 ^ 2} \ sqrt {5} = 2 * 3 \ sqrt { 5} = 6 \ SQRT {5}")  
    我们不能简化素数的平方根，因此最终答案： ![](https://latex.codecogs.com/gif.latex?\sqrt{180}&space;=&space;6\sqrt{5} "\ sqrt {180} = 6 \ sqrt {5}")

### 示例：简化![](https://latex.codecogs.com/gif.latex?\sqrt{1750} "\ SQRT {1750}")

解决步骤：

*   将1750分为素数因子的乘积： ![](https://latex.codecogs.com/gif.latex?1750&space;=&space;2*7*5^3 "1750 = 2 * 7 * 5 ^ 3")
*   使用激进的产品属性并不断分解结果： ![](https://latex.codecogs.com/gif.latex?\sqrt{ab}&space;=&space;\sqrt{a}\sqrt{b}&space;\Rightarrow&space;\sqrt{1750}&space;=&space;\sqrt{2}\sqrt{7}\sqrt{5^3}&space;=&space;\sqrt{2}\sqrt{7}\sqrt{5}\sqrt{5^2}&space;=&space;5\sqrt{2}\sqrt{7}\sqrt{5} "\ sqrt {ab} = \ sqrt {a} \ sqrt {b} \ Rightarrow \ sqrt {1750} = \ sqrt {2} \ sqrt {7} \ sqrt {5 ^ 3} = \ sqrt {2} \ sqrt { 7} \ sqrt {5} \ sqrt {5 ^ 2} = 5 \ sqrt {2} \ sqrt {7} \ sqrt {5}")  
    现在再次使用产品属性来组合不可简化的基础： ![](https://latex.codecogs.com/gif.latex?5\sqrt{2}\sqrt{7}\sqrt{5}&space;=&space;5\sqrt{70} "5 \ sqrt {2} \ sqrt {7} \ sqrt {5} = 5 \ sqrt {70}")

### 术语：

以√49为例：

*   “√”被称为**激进符号** 。
*   49被称为**radicand** 。

#### 更多信息：

*   [Wikihow，简化平方根](https://www.wikihow.com/Simplify-a-Square-Root)
*   [可汗学院](https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-simplify-square-roots/a/simplifying-square-roots-review)