If `x != 0 , (x(x^5)^2)/(x^4)` :

Options

A. `x^5`
B. `x^6`
C. `x^7`
D. `x^8`

Answer

To solve the expression \(\frac{x(x^5)^2}{x^4}\) for \(x \neq 0\), we will use the properties of exponents. Let's break it down step by step.

### Step 1: Simplify the expression in the numerator
The numerator consists of \(x\) multiplied by \((x^5)^2\). We can simplify \((x^5)^2\) using the exponent rule that states \((x^a)^b = x^{a \cdot b}\).

\[ (x^5)^2 = x^{5 \cdot 2} = x^{10} \]

So, the numerator becomes:
\[ x \cdot x^{10} \]

### Step 2: Combine the terms in the numerator
Now we can combine \(x\) and \(x^{10}\) using the property \(x^a \cdot x^b = x^{a + b}\):

\[ x \cdot x^{10} = x^{1 + 10} = x^{11} \]

### Step 3: Rewrite the expression
Now we can rewrite the entire expression:

\[ \frac{x^{11}}{x^4} \]

### Step 4: Apply the division property of exponents
Using the property \(\frac{x^a}{x^b} = x^{a - b}\), we can simplify the expression:

\[ \frac{x^{11}}{x^4} = x^{11 - 4} = x^{7} \]

### Final Answer
Thus, the simplified form of the expression \(\frac{x(x^5)^2}{x^4}\) is:

\[ \boxed{x^7} \]

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