Using binomial theorem, expand each of the following:`(x^(2)-2/x)^(7)`

Answer

To expand the expression \((x^2 - \frac{2}{x})^7\) using the Binomial Theorem, we follow these steps:

### Step 1: Identify the terms in the binomial
The expression can be identified as \(a = x^2\) and \(b = -\frac{2}{x}\). We will use the Binomial Theorem, which states:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Here, \(n = 7\).

### Step 2: Write the expansion using the Binomial Theorem
Using the Binomial Theorem, we can write:
\[ (x^2 - \frac{2}{x})^7 = \sum_{k=0}^{7} \binom{7}{k} (x^2)^{7-k} \left(-\frac{2}{x}\right)^k \]

### Step 3: Simplify each term in the expansion
Now we will simplify each term in the summation:
\[ = \sum_{k=0}^{7} \binom{7}{k} (x^2)^{7-k} (-2)^k \cdot \frac{1}{x^k} \]
This can be rewritten as:
\[ = \sum_{k=0}^{7} \binom{7}{k} (-2)^k x^{2(7-k) - k} \]
\[ = \sum_{k=0}^{7} \binom{7}{k} (-2)^k x^{14 - 3k} \]

### Step 4: Write out the individual terms
Now we will write out the individual terms for \(k = 0\) to \(k = 7\):

- For \(k = 0\):
\[ \binom{7}{0} (-2)^0 x^{14} = 1 \cdot 1 \cdot x^{14} = x^{14} \]

- For \(k = 1\):
\[ \binom{7}{1} (-2)^1 x^{11} = 7 \cdot (-2) \cdot x^{11} = -14x^{11} \]

- For \(k = 2\):
\[ \binom{7}{2} (-2)^2 x^{8} = 21 \cdot 4 \cdot x^{8} = 84x^{8} \]

- For \(k = 3\):
\[ \binom{7}{3} (-2)^3 x^{5} = 35 \cdot (-8) \cdot x^{5} = -280x^{5} \]

- For \(k = 4\):
\[ \binom{7}{4} (-2)^4 x^{2} = 35 \cdot 16 \cdot x^{2} = 560x^{2} \]

- For \(k = 5\):
\[ \binom{7}{5} (-2)^5 x^{-1} = 21 \cdot (-32) \cdot x^{-1} = -672x^{-1} \]

- For \(k = 6\):
\[ \binom{7}{6} (-2)^6 x^{-4} = 7 \cdot 64 \cdot x^{-4} = 448x^{-4} \]

- For \(k = 7\):
\[ \binom{7}{7} (-2)^7 x^{-7} = 1 \cdot (-128) \cdot x^{-7} = -128x^{-7} \]

### Step 5: Combine all terms
Now we combine all the terms:
\[ (x^2 - \frac{2}{x})^7 = x^{14} - 14x^{11} + 84x^{8} - 280x^{5} + 560x^{2} - 672x^{-1} + 448x^{-4} - 128x^{-7} \]

### Final Answer
Thus, the expansion of \((x^2 - \frac{2}{x})^7\) is:
\[ x^{14} - 14x^{11} + 84x^{8} - 280x^{5} + 560x^{2} - 672x^{-1} + 448x^{-4} - 128x^{-7} \]