(2) $$ \sqrt{\frac{1}{8}} $$ (8) $$ \sqrt{-\frac{1}{4}} $$ (8) $$ \sqrt{\frac{2}{10}} $$ (11) $$ \sqrt{\frac{16}{25}}+\sqrt{\frac{9}{25}} $$ (14) $$ \sqrt{\sqrt{\frac{8}{2}}}-\frac{5}{7} $$

Question

UpStudy AI · Accepted Answer

Let's simplify each expression step by step.

1. **For $$ \sqrt{\frac{1}{8}} $$**:
   - We can rewrite this as $$ \sqrt{\frac{1}{2^3}} = \frac{1}{\sqrt{2^3}} = \frac{1}{2\sqrt{2}} $$.

2. **For $$ \sqrt{-\frac{1}{4}} $$**:
   - Since the expression is negative, we can express it as $$ \sqrt{-1} \cdot \sqrt{\frac{1}{4}} = i \cdot \frac{1}{2} = \frac{i}{2} $$.

3. **For $$ \sqrt{\frac{2}{10}} $$**:
   - This simplifies to $$ \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} $$.

4. **For $$ \sqrt{\frac{16}{25}} + \sqrt{\frac{9}{25}} $$**:
   - This simplifies to $$ \frac{4}{5} + \frac{3}{5} = \frac{7}{5} $$.

5. **For $$ \sqrt{\sqrt{\frac{8}{2}}} - \frac{5}{7} $$**:
   - First, simplify $$ \frac{8}{2} = 4 $$, so we have $$ \sqrt{\sqrt{4}} - \frac{5}{7} = \sqrt{2} - \frac{5}{7} $$.

Now, let's compute these expressions using the appropriate tool calls.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{i}{2}$$
- step1: Reduce the fraction:
  $$\frac{1}{2}i$$
Calculate or simplify the expression $$ \frac{1}{2\sqrt(2)} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{2\sqrt{2}}$$
- step1: Multiply by the Conjugate:
  $$\frac{\sqrt{2}}{2\sqrt{2}\times \sqrt{2}}$$
- step2: Multiply the numbers:
  $$\frac{\sqrt{2}}{4}$$
Calculate or simplify the expression $$ \sqrt(2) - \frac{5}{7} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{2}-\frac{5}{7}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{\sqrt{2}\times 7}{7}-\frac{5}{7}$$
- step2: Transform the expression:
  $$\frac{\sqrt{2}\times 7-5}{7}$$
- step3: Reorder the terms:
  $$\frac{7\sqrt{2}-5}{7}$$
Calculate or simplify the expression $$ \frac{1}{\sqrt(5)} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{\sqrt{5}}$$
- step1: Multiply by the Conjugate:
  $$\frac{\sqrt{5}}{\sqrt{5}\times \sqrt{5}}$$
- step2: Multiply the numbers:
  $$\frac{\sqrt{5}}{5}$$
Calculate or simplify the expression $$ \frac{4}{5} + \frac{3}{5} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{4}{5}+\frac{3}{5}$$
- step1: Transform the expression:
  $$\frac{4+3}{5}$$
- step2: Add the numbers:
  $$\frac{7}{5}$$
Here are the simplified results for each expression:

1. **For $$ \sqrt{\frac{1}{8}} $$**:
   $$
   \sqrt{\frac{1}{8}} = \frac{\sqrt{2}}{4}
   $$

2. **For $$ \sqrt{-\frac{1}{4}} $$**:
   $$
   \sqrt{-\frac{1}{4}} = \frac{1}{2}i
   $$

3. **For $$ \sqrt{\frac{2}{10}} $$**:
   $$
   \sqrt{\frac{2}{10}} = \frac{\sqrt{5}}{5}
   $$

4. **For $$ \sqrt{\frac{16}{25}} + \sqrt{\frac{9}{25}} $$**:
   $$
   \sqrt{\frac{16}{25}} + \sqrt{\frac{9}{25}} = \frac{7}{5}
   $$

5. **For $$ \sqrt{\sqrt{\frac{8}{2}}} - \frac{5}{7} $$**:
   $$
   \sqrt{\sqrt{\frac{8}{2}}} - \frac{5}{7} = \frac{7\sqrt{2}-5}{7}
   $$

These are the simplified forms of the given expressions.
Here are the simplified results for each expression: 1. $$ \sqrt{\frac{1}{8}} = \frac{\sqrt{2}}{4} $$ 2. $$ \sqrt{-\frac{1}{4}} = \frac{1}{2}i $$ 3. $$ \sqrt{\frac{2}{10}} = \frac{\sqrt{5}}{5} $$ 4. $$ \sqrt{\frac{16}{25}} + \sqrt{\frac{9}{25}} = \frac{7}{5} $$ 5. $$ \sqrt{\sqrt{\frac{8}{2}}} - \frac{5}{7} = \frac{7\sqrt{2} - 5}{7} $$

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