Let's work through each term in the expression [tex]\(3 \sqrt{\frac{7}{25}} - \sqrt{\frac{28}{25}} + \sqrt{\frac{63}{25}}\)[/tex] step by step.
### Step 1: Simplify each term
1. First Term: [tex]\(3 \sqrt{\frac{7}{25}}\)[/tex]
- We can simplify inside the square root:
[tex]\[
\sqrt{\frac{7}{25}} = \frac{\sqrt{7}}{\sqrt{25}} = \frac{\sqrt{7}}{5}
\][/tex]
- Therefore:
[tex]\[
3 \sqrt{\frac{7}{25}} = 3 \cdot \frac{\sqrt{7}}{5} = \frac{3\sqrt{7}}{5}
\][/tex]
2. Second Term: [tex]\(\sqrt{\frac{28}{25}}\)[/tex]
- We again simplify inside the square root:
[tex]\[
\sqrt{\frac{28}{25}} = \frac{\sqrt{28}}{\sqrt{25}} = \frac{\sqrt{28}}{5}
\][/tex]
- Simplify [tex]\(\sqrt{28}\)[/tex]:
[tex]\[
\sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}
\][/tex]
- So:
[tex]\[
\sqrt{\frac{28}{25}} = \frac{2\sqrt{7}}{5}
\][/tex]
3. Third Term: [tex]\(\sqrt{\frac{63}{25}}\)[/tex]
- Similarly simplify inside the square root:
[tex]\[
\sqrt{\frac{63}{25}} = \frac{\sqrt{63}}{\sqrt{25}} = \frac{\sqrt{63}}{5}
\][/tex]
- Simplify [tex]\(\sqrt{63}\)[/tex]:
[tex]\[
\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}
\][/tex]
- So:
[tex]\[
\sqrt{\frac{63}{25}} = \frac{3\sqrt{7}}{5}
\][/tex]
### Step 2: Sum up the terms
Now, we sum the simplified terms:
[tex]\[
3 \sqrt{\frac{7}{25}} - \sqrt{\frac{28}{25}} + \sqrt{\frac{63}{25}} = \frac{3\sqrt{7}}{5} - \frac{2\sqrt{7}}{5} + \frac{3\sqrt{7}}{5}
\][/tex]
Combine the fractions:
[tex]\[
\frac{3\sqrt{7}}{5} - \frac{2\sqrt{7}}{5} + \frac{3\sqrt{7}}{5} = \frac{3\sqrt{7} - 2\sqrt{7} + 3\sqrt{7}}{5} = \frac{4\sqrt{7}}{5}
\][/tex]
### Step 3: Rationalize and simplify
The numerical value of [tex]\(\frac{4\sqrt{7}}{5}\)[/tex] rounds to [tex]\(2.11660105\)[/tex].
Next, we rationalize the expression to match the given options:
[tex]\[
\frac{4\sqrt{7}}{5}
\][/tex]
We recognize that [tex]\(\sqrt{7}\)[/tex] can be seen in the multiple-choice answers.
Finally, we find:
[tex]\[
\left(\frac{4}{5}\right) \sqrt{7} = \frac{4}{5} \cdot \sqrt{7}
\][/tex]
Given [tex]\(\sqrt{7}\)[/tex] as a radical:
[tex]\[
\frac{4}{5} = \frac{8}{10} = 0.8
\][/tex]
Thus:
[tex]\[
\frac{4}{5}\sqrt{7} = 0.8\sqrt{7}
\][/tex]
Among the answer options, the correct one that simplifies to [tex]\(0.8\sqrt{7}\)[/tex] is:
[tex]\[
\frac{8}{10} \sqrt{7} = 0.8 \sqrt{7}
\][/tex]
So the correct choice from the given answers is:
[tex]\[
\boxed{ 0.8 \sqrt{7} }
\][/tex]
