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

Answer

To solve the expression \(\frac{(1-2x)^{1/3} + (1+5x)^{-3/2}}{(9+x)^{1/2}}\) for small values of \(x\), we can use the binomial approximation. Let's break it down step by step.

### Step 1: Simplify the numerator
We start with the numerator:

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

Using the binomial expansion for small \(x\):
- For \((1-2x)^{1/3}\):
\[ (1-2x)^{1/3} \approx 1 + \frac{1}{3}(-2x) = 1 - \frac{2}{3}x \]

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

Now, we can combine these results:
\[ (1-2x)^{1/3} + (1+5x)^{-3/2} \approx \left(1 - \frac{2}{3}x\right) + \left(1 - \frac{15}{2}x\right) \]
\[ = 2 - \left(\frac{2}{3} + \frac{15}{2}\right)x \]

To combine the coefficients of \(x\), we need a common denominator. The least common multiple of 3 and 2 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, the numerator simplifies to:
\[ 2 - \frac{49}{6}x \]

### Step 2: Simplify the denominator
Now, we simplify the denominator:
\[ (9+x)^{1/2} \]
Using the binomial expansion:
\[ (9+x)^{1/2} = 3\left(1 + \frac{x}{9}\right)^{1/2} \approx 3\left(1 + \frac{1}{2}\frac{x}{9}\right) = 3 + \frac{x}{6} \]

### Step 3: Combine the results
Now we have:
\[ \frac{2 - \frac{49}{6}x}{3 + \frac{x}{6}} \]

To simplify this expression, 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 4: Evaluate for small \(x\)
For small \(x\), we can ignore the \(x\) in the denominator:
\[ \approx \frac{12 - 49x}{18} \approx \frac{12}{18} - \frac{49}{18}x \]
\[ = \frac{2}{3} - \frac{49}{18}x \]

### Final Result
Thus, the value of the expression for small \(x\) is approximately:
\[ \frac{2}{3} - \frac{49}{18}x \]