3. Simplify the surd: $$ \sqrt{200} $$ Evaluate: $$ \sqrt[8]{63^{8}} $$

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UpStudy AI · Accepted Answer

**Simplify the surd $$ \sqrt{200} $$:**

1. **Factorize 200:**  
   We note that  
   $$
   200 = 100 \times 2
   $$

2. **Separate the square root:**  
   Using the property $$\sqrt{ab} = \sqrt{a}\sqrt{b}$$, we have  
   $$
   \sqrt{200} = \sqrt{100} \times \sqrt{2}
   $$

3. **Simplify $$\sqrt{100}$$:**  
   Since $$\sqrt{100} = 10$$, it follows that  
   $$
   \sqrt{200} = 10\sqrt{2}
   $$

---

**Evaluate $$ \sqrt[8]{63^{8}} $$:**

1. **Use the property of roots and exponents:**  
   In general, for a non-negative real number $$a$$ and an integer $$n$$,  
   $$
   \sqrt[n]{a^n} = a
   $$  
   (more precisely, it equals $$|a|$$ – since $$63$$ is positive, we have $$ |63| = 63 $$).

2. **Apply to the given expression:**  
   Therefore,  
   $$
   \sqrt[8]{63^{8}} = 63
   $$

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**Final Answers:**

- $$ \sqrt{200} = 10\sqrt{2} $$  
- $$ \sqrt[8]{63^{8}} = 63 $$
$$ \sqrt{200} = 10\sqrt{2} $$ and $$ \sqrt[8]{63^{8}} = 63 $$.

Simplify the surd:
Evaluate:

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