`[(root(7)(x^((-7)/3)))^(-3/2)]^(10)=?`

Options

A. `x^(5)`
B. `x^(-5)`
C. x
D. `1/x`

Answer

To simplify the expression \([( \sqrt[7]{x^{(-7/3)}} )^{(-3/2)}]^{10}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ [( \sqrt[7]{x^{(-7/3)}} )^{(-3/2)}]^{10} \]
The 7th root can be expressed as a power of \( \frac{1}{7} \):
\[ = [(x^{(-7/3)})^{(1/7)}]^{(-3/2)}]^{10} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ = [x^{(-7/3) \cdot (1/7)}]^{(-3/2)}]^{10} \]

### Step 3: Simplify the exponent
Now we simplify the exponent:
\[ = [x^{(-1/3)}]^{(-3/2)}]^{10} \]

### Step 4: Apply the power of a power rule again
Now we apply the power of a power rule again:
\[ = x^{(-1/3) \cdot (-3/2) \cdot 10} \]

### Step 5: Calculate the exponent
Calculating the exponent:
\[ = x^{(1/3) \cdot (3/2) \cdot 10} \]
\[ = x^{(1 \cdot 10)/2} \]
\[ = x^{5} \]

### Final Answer
Thus, the simplified expression is:
\[ x^{5} \]

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