If x be very small compered with such that
`(sqrt(1+ x)+root(3)((1 - x)^(2)))/(sqrt(1+x)+(1+x))`~~a bx`, then the values of a and b are

Options

A. `A= 1, b = (5)/(6)`
B. `a = 1, b = - (5)/(6)`
C. `a = 1, b = (5)/(6)`
D. `a = 1, b = - (5 )/(3)`

Explanation

To solve the given problem, we need to simplify the expression:

\[ \frac{\sqrt{1+x} + 3\sqrt[3]{(1-x)^2}}{\sqrt{1+x} + (1+x)} \]

and express it in the form \( a - bx \) where \( x \) is very small.

### Step-by-step Solution:

1. **Expand the terms using Binomial Theorem:**
- For \( \sqrt{1+x} \), we can use the binomial expansion:
\[ \sqrt{1+x} \approx 1 + \frac{1}{2}x \quad \text{(for small } x\text{)} \]
- For \( \sqrt[3]{(1-x)^2} \), we first expand \( (1-x)^2 \):
\[ (1-x)^2 = 1 - 2x + x^2 \]
Now, apply the binomial expansion:
\[ \sqrt[3]{1 - 2x + x^2} \approx 1 - \frac{2}{3}x \quad \text{(neglecting higher order terms)} \]

2. **Substituting back into the expression:**
- Now substitute these expansions into the original expression:
\[ \sqrt{1+x} + 3\sqrt[3]{(1-x)^2} \approx \left(1 + \frac{1}{2}x\right) + 3\left(1 - \frac{2}{3}x\right) \]
Simplifying this gives:
\[ 1 + \frac{1}{2}x + 3 - 2x = 4 - \frac{3}{2}x \]

3. **Expand the denominator:**
- For the denominator \( \sqrt{1+x} + (1+x) \):
\[ \sqrt{1+x} + (1+x) \approx \left(1 + \frac{1}{2}x\right) + (1+x) = 2 + \frac{3}{2}x \]

4. **Combine the numerator and denominator:**
- Now we have:
\[ \frac{4 - \frac{3}{2}x}{2 + \frac{3}{2}x} \]

5. **Simplify the fraction:**
- To simplify, we can multiply the numerator and denominator by \( \frac{1}{2} \):
\[ \frac{2 - \frac{3}{4}x}{1 + \frac{3}{4}x} \]
- Now, apply the binomial expansion to the denominator:
\[ \frac{1}{1 + \frac{3}{4}x} \approx 1 - \frac{3}{4}x \quad \text{(for small } x\text{)} \]
- Thus:
\[ \frac{2 - \frac{3}{4}x}{1 + \frac{3}{4}x} \approx (2 - \frac{3}{4}x)(1 - \frac{3}{4}x) = 2 - \frac{3}{4}x - \frac{3}{2}x + O(x^2) \]
- This simplifies to:
\[ 2 - \frac{9}{4}x \]

6. **Final expression:**
- We need to express this in the form \( a - bx \):
\[ 2 - \frac{9}{4}x \approx a - bx \]
- Thus, we have:
\[ a = 2 \quad \text{and} \quad b = \frac{9}{4} \]

### Conclusion:
The values of \( a \) and \( b \) are:
\[ a = 2, \quad b = \frac{9}{4} \]