**Find the first three terms in the expansion of (9−16x)^3/2​ in ascending powers of *x*, and state the set of values for which this expansion is valid.**

To find the first three terms in the expansion of \((9 - 16x)^{\frac{3}{2}}\) in ascending powers of \(x\), we can use the binomial series expansion. The binomial series expansion for \((1 + u)^n\) is given by:

\[
(1 + u)^n = 1 + nu + \frac{n(n-1)}{2!}u^2 + \frac{n(n-1)(n-2)}{3!}u^3 + \dots
\]

First, rewrite the expression \((9 - 16x)^{\frac{3}{2}}\) in a form that allows us to apply the binomial series:

\[
(9 - 16x)^{\frac{3}{2}} = 9^{\frac{3}{2}} \left(1 - \frac{16x}{9}\right)^{\frac{3}{2}}
\]

Calculate \(9^{\frac{3}{2}}\):

\[
9^{\frac{3}{2}} = \left(3^2\right)^{\frac{3}{2}} = 3^3 = 27
\]

Thus, the expression becomes:

\[
27\left(1 - \frac{16x}{9}\right)^{\frac{3}{2}}
\]

Now, apply the binomial series expansion to \(\left(1 - \frac{16x}{9}\right)^{\frac{3}{2}}\) with \(u = -\frac{16x}{9}\) and \(n = \frac{3}{2}\):

\[
\left(1 - \frac{16x}{9}\right)^{\frac{3}{2}} = 1 + \frac{3}{2}\left(-\frac{16x}{9}\right) + \frac{\frac{3}{2}\left(\frac{3}{2}-1\right)}{2!}\left(-\frac{16x}{9}\right)^2 + \dots
\]

Simplify the first three terms:

\[
1 + \frac{3}{2} \cdot \left(-\frac{16x}{9}\right) + \frac{\frac{3}{2} \cdot \frac{1}{2}}{2} \left(-\frac{16x}{9}\right)^2
\]

\[
= 1 - \frac{24x}{9} + \frac{3 \cdot 1 \cdot 256x^2}{8 \cdot 81}
\]

\[
= 1 - \frac{8x}{3} + \frac{256x^2}{162}
\]

\[
= 1 - \frac{8x}{3} + \frac{128x^2}{81}
\]

Now, multiply these by 27:

\[
27 \left(1 - \frac{8x}{3} + \frac{128x^2}{81}\right) = 27 - 72x + \frac{3456x^2}{81}
\]

\[
= 27 - 72x +  \frac{1152x^2}{27} 
= 27 - 72x +  \frac{128x^2}{3} 
\]

### Therefore, the first three terms in the expansion are:
\[
27 - 72x + \frac{128x^2}{3}
\]

### Validity of the expansion:
The binomial expansion is valid when \(|u| < 1\), meaning:

\[
\left|\frac{-16x}{9}\right| < 1
\]

\[
\left|x\right| < \frac{9}{16}
\]

### So, the expansion is valid for:
\[
|x| < \frac{9}{16}
\]

Would you like further details or have any questions?  

Here are 5 related questions:
1. What are the next three terms in the expansion of \((9 - 16x)^{\frac{3}{2}}\)?
2. How would you find the radius of convergence for a series expansion?
3. Can you expand \((1 - 2x)^{-\frac{1}{2}}\) using the binomial series?
4. What is the binomial expansion for \((1 + x)^n\) when \(n\) is a negative integer?
5. How can the binomial theorem be applied to approximate functions near a point?

**Tip:** The binomial series can be used to approximate functions when the exponent is not an integer.

Find the first three terms in the expansion of (9−16x)^(3/2) in ascending powers of x, and state the set of values for which this expansion is valid.

Discover how to find the first three terms in the expansion of (9−16x)^(3/2) using the binomial theorem, and learn the range of validity for this expansion. This problem involves key algebraic concepts, binomial expansion, and series analysis. Suitable for advanced high school or undergraduate students.

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