If `x` is so small such that its square and higher powers may be neglected, the find the value of
`((1-2x)^(1//3)+(1+5x)^(-3//2))/((9+x)^(1//2))`

Answer

To solve the expression

\[ \frac{(1-2x)^{1/3} + (1+5x)^{-3/2}}{(9+x)^{1/2}}, \]

we will use the binomial expansion for small values of \(x\). Since \(x\) is small, we can neglect \(x^2\) and higher powers.

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

Using the binomial expansion, we have:

\[ (1 + u)^n \approx 1 + nu \quad \text{for small } u. \]

Here, \(u = -2x\) and \(n = \frac{1}{3}\):

\[ (1-2x)^{1/3} \approx 1 + \frac{1}{3}(-2x) = 1 - \frac{2}{3}x. \]

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

Similarly, we can expand \((1 + 5x)^{-3/2}\) using \(u = 5x\) and \(n = -\frac{3}{2}\):

\[ (1 + 5x)^{-3/2} \approx 1 - \frac{3}{2}(5x) = 1 - \frac{15}{2}x. \]

### Step 3: Combine the expansions

Now we can combine the results from Steps 1 and 2:

\[ (1-2x)^{1/3} + (1+5x)^{-3/2} \approx \left(1 - \frac{2}{3}x\right) + \left(1 - \frac{15}{2}x\right). \]

Combining these gives:

\[ 2 - \left(\frac{2}{3} + \frac{15}{2}\right)x. \]

To combine the coefficients of \(x\), we need a common denominator (which is 6):

\[ \frac{2}{3} = \frac{4}{6}, \quad \frac{15}{2} = \frac{45}{6}. \]

Thus,

\[ \frac{2}{3} + \frac{15}{2} = \frac{4}{6} + \frac{45}{6} = \frac{49}{6}. \]

So we have:

\[ (1-2x)^{1/3} + (1+5x)^{-3/2} \approx 2 - \frac{49}{6}x. \]

### Step 4: Expand \((9+x)^{1/2}\)

Next, we expand \((9+x)^{1/2}\):

\[ (9+x)^{1/2} = 9^{1/2}\left(1 + \frac{x}{9}\right)^{1/2} \approx 3\left(1 + \frac{1}{2}\cdot\frac{x}{9}\right) = 3 + \frac{x}{6}. \]

### Step 5: Substitute into the original expression

Now we substitute our expansions back into the original expression:

\[ \frac{2 - \frac{49}{6}x}{3 + \frac{x}{6}}. \]

### Step 6: Simplify the expression

To simplify, we can multiply the numerator and denominator by 6 to eliminate the fractions:

\[ \frac{6(2 - \frac{49}{6}x)}{6(3 + \frac{x}{6})} = \frac{12 - 49x}{18 + x}. \]

### Step 7: Evaluate the limit as \(x \to 0\)

As \(x\) approaches 0, we can find the value of the expression:

\[ \frac{12 - 49(0)}{18 + 0} = \frac{12}{18} = \frac{2}{3}. \]

### Final Answer

Thus, the value of the expression is

\[ \frac{2}{3}. \]