If |x| is so small that all terms containing `x^2` and higher powers of x can be neglected , then the approximate value of `((3 - 5x)^(1//2))/((5 - 3x)^2), ` where `x = (1)/(sqrt363)` , is

Options

A. `(sqrt3)/(25)`
B. `(1+30sqrt3)/(75)`
C. `(1-30sqrt3)/(75)`
D. `(1+30sqrt3)/(750)`

Answer

To solve the problem, we need to find the approximate value of the expression:

\[ \frac{(3 - 5x)^{1/2}}{(5 - 3x)^2} \]

given that \( x = \frac{1}{\sqrt{363}} \) and neglecting all terms containing \( x^2 \) and higher powers of \( x \).

### Step 1: Rewrite the expression

We start by rewriting the expression:

\[ \frac{(3 - 5x)^{1/2}}{(5 - 3x)^2} = (3 - 5x)^{1/2} \cdot (5 - 3x)^{-2} \]

### Step 2: Expand \( (3 - 5x)^{1/2} \)

Using the binomial expansion for \( (1 + u)^n \) where \( u = -\frac{5x}{3} \) and \( n = \frac{1}{2} \):

\[ (3 - 5x)^{1/2} = 3^{1/2} \left(1 - \frac{5x}{3}\right)^{1/2} \approx \sqrt{3} \left(1 - \frac{5}{6}x\right) \]

This approximation holds because we neglect terms involving \( x^2 \) and higher.

### Step 3: Expand \( (5 - 3x)^{-2} \)

Similarly, for \( (5 - 3x)^{-2} \):

\[ (5 - 3x)^{-2} = 5^{-2} \left(1 - \frac{3x}{5}\right)^{-2} \approx \frac{1}{25} \left(1 + \frac{6}{5}x\right) \]

### Step 4: Combine the expansions

Now we combine the two expansions:

\[ (3 - 5x)^{1/2} \cdot (5 - 3x)^{-2} \approx \sqrt{3} \left(1 - \frac{5}{6}x\right) \cdot \frac{1}{25} \left(1 + \frac{6}{5}x\right) \]

### Step 5: Simplify the expression

Now we simplify:

\[ \approx \frac{\sqrt{3}}{25} \left(1 - \frac{5}{6}x + \frac{6}{5}x\right) \]

Combining the terms in the bracket:

\[ -\frac{5}{6}x + \frac{6}{5}x = \left(-\frac{25}{30} + \frac{36}{30}\right)x = \frac{11}{30}x \]

Thus, we have:

\[ \approx \frac{\sqrt{3}}{25} \left(1 + \frac{11}{30}x\right) \]

### Step 6: Substitute \( x \)

Now substituting \( x = \frac{1}{\sqrt{363}} \):

\[ \approx \frac{\sqrt{3}}{25} \left(1 + \frac{11}{30} \cdot \frac{1}{\sqrt{363}}\right) \]

### Step 7: Final expression

This gives us the final approximate value:

\[ \approx \frac{\sqrt{3}}{25} + \frac{11\sqrt{3}}{750\sqrt{363}} \]

### Conclusion

Thus, the approximate value of the expression is:

\[ \frac{\sqrt{3}}{25} + \frac{11\sqrt{3}}{750\sqrt{363}} \]