Write down the first three terms is the following expansions
`(8 - 5x)^(2//3)`

Answer

To find the first three terms of the expansion of \((8 - 5x)^{\frac{2}{3}}\), we can use the Binomial Theorem, which states that:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

In our case, we can rewrite the expression as:

\[ (8 - 5x)^{\frac{2}{3}} = (8(1 - \frac{5x}{8}))^{\frac{2}{3}} \]

Now, we can set \(a = 8\) and \(b = -\frac{5x}{8}\) and \(n = \frac{2}{3}\).

### Step 1: Identify the first three terms
Using the Binomial expansion, we can find the first three terms by calculating for \(k = 0\), \(k = 1\), and \(k = 2\).

### Step 2: Calculate for \(k = 0\)
\[ \text{Term for } k=0: \binom{\frac{2}{3}}{0} \cdot 8^{\frac{2}{3}} \cdot \left(-\frac{5x}{8}\right)^0 = 1 \cdot 8^{\frac{2}{3}} \cdot 1 = 8^{\frac{2}{3}} = 4 \]

### Step 3: Calculate for \(k = 1\)
\[ \text{Term for } k=1: \binom{\frac{2}{3}}{1} \cdot 8^{\frac{2}{3}-1} \cdot \left(-\frac{5x}{8}\right)^1 = \frac{2}{3} \cdot 8^{\frac{2}{3}-1} \cdot \left(-\frac{5x}{8}\right) \]
Calculating \(8^{\frac{2}{3}-1} = 8^{-\frac{1}{3}} = \frac{1}{2}\):
\[ = \frac{2}{3} \cdot \frac{1}{2} \cdot \left(-\frac{5x}{8}\right) = -\frac{5x}{24} \]

### Step 4: Calculate for \(k = 2\)
\[ \text{Term for } k=2: \binom{\frac{2}{3}}{2} \cdot 8^{\frac{2}{3}-2} \cdot \left(-\frac{5x}{8}\right)^2 = \frac{\frac{2}{3} \cdot \frac{-1}{3}}{2} \cdot 8^{-\frac{4}{3}} \cdot \left(\frac{25x^2}{64}\right) \]
Calculating \(8^{-\frac{4}{3}} = \frac{1}{16}\):
\[ = \frac{-1}{9} \cdot \frac{1}{16} \cdot \frac{25x^2}{64} = -\frac{25x^2}{864} \]

### Final Result
Now, we can combine the first three terms:
\[ 4 - \frac{5x}{24} - \frac{25x^2}{864} \]

Thus, the first three terms of the expansion of \((8 - 5x)^{\frac{2}{3}}\) are:
\[ 4 - \frac{5x}{24} - \frac{25x^2}{864} \]