Question

Math

Posted 4 months ago

```
1. 化简: $\sqrt{6-3 \sqrt{3}}+\sqrt{6+3 \sqrt{3}}$ 的结果是 ( )
A. 6
B. $\sqrt{6}$
C. $3 \sqrt{3}$
D. $3 \sqrt{2}$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

We can rewrite the terms as $\sqrt{3(2-\sqrt{3})} + \sqrt{3(2+\sqrt{3})}$

step 3

Factoring out the common term $\sqrt{3}$, we get $\sqrt{3}(\sqrt{2-\sqrt{3}} + \sqrt{2+\sqrt{3}})$

step 4

The expression simplifies to $3\sqrt{2}$, as the terms inside the parentheses combine to form a conjugate pair, which results in the simplification of the square roots

D

Key Concept

Conjugate Pairs

Explanation

When simplifying square roots of the form $\sqrt{a+b}$ and $\sqrt{a-b}$, where $a$ and $b$ are positive real numbers, the sum of these roots can often be simplified by recognizing them as conjugate pairs. This allows for the simplification of the square roots and often results in a rational number.

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