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

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

given that \( x \) is very small, we can use the binomial expansion for small values of \( x \).

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

For this term, we have \( u = 5x \) and \( n = -\frac{3}{2} \):

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

### Step 3: Combine the expansions

Now we can combine the two expansions:

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

Combining the constant terms and the \( x \) terms:

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

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 \( (9+x)^{1/2} \)

Now, for the denominator \( (9+x)^{1/2} \):

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

### Step 5: Form the final expression

Now we can substitute back into our original expression:

\[ \frac{(1-2x)^{1/3} + (1+5x)^{-3/2}}{(9+x)^{1/2}} \approx \frac{2 - \frac{49}{6}x}{3 + \frac{x}{6}} \]

### Step 6: Simplify the expression

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

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

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

As \( x \) approaches 0, the expression simplifies to:

\[ \frac{12}{18} = \frac{2}{3} \]

### Final Answer

Thus, the value of the expression is:

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