solve it \sqrt{(\frac{145}{37} - \frac{283}{37})^2 + (\frac{298}{37} - \frac{160}{37})^2 + (\frac{226}{37} - \frac{295}{37})^2}

The given expression is:

\[
\sqrt{\left(\frac{145}{37} - \frac{283}{37}\right)^2 + \left(\frac{298}{37} - \frac{160}{37}\right)^2 + \left(\frac{226}{37} - \frac{295}{37}\right)^2}
\]

Let's simplify step by step.

### Step 1: Simplify the differences inside the square roots

1. \(\frac{145}{37} - \frac{283}{37} = \frac{145 - 283}{37} = \frac{-138}{37}\)
2. \(\frac{298}{37} - \frac{160}{37} = \frac{298 - 160}{37} = \frac{138}{37}\)
3. \(\frac{226}{37} - \frac{295}{37} = \frac{226 - 295}{37} = \frac{-69}{37}\)

### Step 2: Square each term

1. \(\left(\frac{-138}{37}\right)^2 = \frac{138^2}{37^2} = \frac{19044}{1369}\)
2. \(\left(\frac{138}{37}\right)^2 = \frac{138^2}{37^2} = \frac{19044}{1369}\)
3. \(\left(\frac{-69}{37}\right)^2 = \frac{69^2}{37^2} = \frac{4761}{1369}\)

### Step 3: Add the squared terms

\[
\frac{19044}{1369} + \frac{19044}{1369} + \frac{4761}{1369} = \frac{19044 + 19044 + 4761}{1369} = \frac{42849}{1369}
\]

### Step 4: Take the square root

\[
\sqrt{\frac{42849}{1369}} = \frac{\sqrt{42849}}{\sqrt{1369}} = \frac{207}{37}
\]

Thus, the value of the expression is:

\[
\boxed{\frac{207}{37}}
\]

Would you like further details or have any questions?

### 5 Related Questions:
1. How do you simplify fractions in algebraic expressions?
2. What are the properties of square roots when simplifying terms?
3. How do you handle subtraction of fractions with like denominators?
4. What is the process for squaring a fraction?
5. How do you add and subtract algebraic fractions efficiently?

### Tip:
Always remember to factor out common denominators when dealing with multiple terms under a square root—it makes simplification much easier!

Solve and simplify the expression involving fractions and square roots: \sqrt{(\frac{145}{37} - \frac{283}{37})^2 + (\frac{298}{37} - \frac{160}{37})^2 + (\frac{226}{37} - \frac{295}{37})^2}. This problem breaks down algebraic concepts like fraction operations and square roots, suitable for students in Grades 9-10.

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